algebra 1 problem solving

*BRAND NEW* Self-paced Introduction to Counting & Probability.   Enroll   today!

Need help finding the right class? Have a question about how classes work?

This is the first AoPS course I've ever taken, and I absolutely loved it. Definitely will be taking more classes here. This is the best math program I've ever had.  It's very challenging but also very fun.

Something appears to not have loaded correctly.

Click to refresh .

• Inspiration

IXL offers hundreds of Algebra 1 skills to explore and learn! Not sure where to start? Go to your personalized Recommendations wall to find a skill that looks interesting, or select a skill plan that aligns to your textbook, state standards, or standardized test.

A. Numbers and operations

• 1 Add, subtract, multiply, and divide integers
• 2 Evaluate numerical expressions involving integers
• 3 Convert between decimals and fractions
• 4 Add and subtract rational numbers
• 5 Multiply and divide rational numbers
• 6 Simplify complex fractions
• 7 Evaluate numerical expressions involving rational numbers
• 8 Square roots
• 9 Cube roots
• 10 Sort rational and irrational numbers
• 11 Classify rational and irrational numbers
• 12 Properties of operations on rational and irrational numbers
• 13 Classify numbers

B. Expressions

• 1 Write variable expressions
• 2 Evaluate variable expressions involving integers
• 3 Evaluate variable expressions involving rational numbers
• 4 Evaluate rational expressions
• 5 Properties of addition and multiplication
• 6 Distributive property
• 7 Simplify linear expressions using properties
• 8 Sort factors of variable expressions
• 9 Simplify variable expressions involving like terms and the distributive property
• 10 Identify equivalent linear expressions

C. Solve linear equations

• 1 Write variable equations
• 2 Does x satisfy the equation?
• 3 Which x satisfies an equation?
• 4 Properties of equality
• 5 Identify equivalent equations
• 6 Model and solve linear equations using algebra tiles
• 7 Solve one-step linear equations
• 8 Solve two-step linear equations
• 9 Solve one-step and two-step linear equations: word problems
• 10 Solve multi-step linear equations
• 11 Consecutive integer problems
• 12 Solve linear equations with variables on both sides
• 13 Solve linear equations: complete the solution
• 14 Find the number of solutions to a linear equation
• 15 Create linear equations with no solutions or infinitely many solutions
• 16 Solve linear equations with variables on both sides: word problems
• 17 Solve linear equations: mixed review
• 18 Rearrange multi-variable equations

D. Geometry and measurement

• 1 Area and perimeter: word problems
• 3 Surface area
• 4 Precision
• 5 Greatest possible error
• 6 Minimum and maximum area and volume
• 7 Percent error
• 8 Percent error: area and volume

E. Problem solving with units

• 1 Scale drawings: word problems
• 2 Convert rates and measurements: customary units
• 3 Convert rates and measurements: metric units
• 4 Unit prices with unit conversions
• 5 Multi-step problems with unit conversions
• 6 Rate of travel: word problems
• 7 Weighted averages: word problems

F. Solve linear inequalities

• 1 Graph inequalities
• 2 Write inequalities from graphs
• 3 Identify solutions to inequalities
• 4 Solve one-step linear inequalities: addition and subtraction
• 5 Solve one-step linear inequalities: multiplication and division
• 6 Solve one-step linear inequalities
• 7 Graph solutions to one-step linear inequalities
• 8 Solve two-step linear inequalities
• 9 Graph solutions to two-step linear inequalities
• 10 Solve advanced linear inequalities
• 11 Graph solutions to advanced linear inequalities
• 12 Graph compound inequalities
• 13 Write compound inequalities from graphs
• 14 Solve compound inequalities
• 15 Graph solutions to compound inequalities
• 1 Sets, subsets, and complements
• 2 Set-builder notation
• 3 Interval notation
• 4 Unions and intersections of sets

H. Absolute value equations and inequalities

• 1 Solve absolute value equations
• 2 Graph solutions to absolute value equations
• 3 Write absolute value equations from graphs
• 4 Solve absolute value inequalities
• 5 Graph solutions to absolute value inequalities

I. Coordinate plane

• 1 Coordinate plane review
• 2 Midpoint formula: find the midpoint
• 3 Midpoint formula: find the endpoint
• 4 Distance between two points

J. Direct and inverse variation

• 1 Find the constant of variation
• 2 Write direct variation equations
• 3 Write and solve direct variation equations
• 4 Identify direct variation and inverse variation
• 5 Write inverse variation equations
• 6 Write and solve inverse variation equations
• 1 Find the slope of a graph
• 2 Find the slope from two points
• 3 Find the slope from a table
• 4 Find a missing coordinate using slope

L. Linear equations

• 1 Does (x, y) satisfy the linear equation?
• 2 Relate the graph of a linear equation to its solutions
• 3 Slope-intercept form: find the slope and y-intercept
• 4 Slope-intercept form: graph an equation
• 5 Slope-intercept form: write an equation from a graph
• 6 Slope-intercept form: write an equation
• 7 Slope-intercept form: write an equation from a table
• 8 Linear equations: solve for y
• 9 Write linear equations in standard form
• 10 Standard form: find x- and y-intercepts
• 11 Standard form: graph a line from an equation
• 12 Equations of horizontal and vertical lines
• 13 Graph a horizontal or vertical line
• 14 Point-slope form: graph an equation
• 15 Point-slope form: write an equation
• 16 Point-slope form: write an equation from a graph
• 17 Find the slope and intercepts from an equation
• 18 Slopes of parallel and perpendicular lines
• 19 Write an equation for a parallel or perpendicular line

M. Function concepts

• 1 Relations: convert between tables, graphs, mappings, and lists of points
• 2 Domain and range of relations
• 3 Identify independent and dependent variables
• 4 Identify functions
• 5 Identify functions: vertical line test
• 6 Find values using function graphs
• 7 Evaluate a function
• 8 Evaluate a function: plug in an expression
• 9 Complete a function table from a graph
• 10 Complete a function table from an equation
• 11 Find solutions using a table
• 12 Approximate solutions using a table
• 13 Interpret functions using everyday language
• 14 Identify graphs: word problems
• 15 Rate of change: tables
• 16 Rate of change: graphs

N. Linear functions

• 1 Identify linear functions from graphs and equations
• 2 Identify linear functions from tables
• 3 Complete a table and graph a linear function
• 4 Evaluate a linear function from its graph: word problems
• 5 Interpret the slope and y-intercept of a linear function
• 6 Write a linear function: word problems
• 7 Domain and range of linear functions: graphs
• 8 Domain and range of linear functions: word problems
• 9 Compare linear functions: graphs and equations
• 10 Compare linear functions: tables, graphs, and equations
• 11 Transformations of linear functions

O. Systems of linear equations

• 1 Is (x, y) a solution to the system of equations?
• 2 Solve a system of equations by graphing
• 3 Solve a system of equations by graphing: word problems
• 4 Find the number of solutions to a system of equations by graphing
• 5 Find the number of solutions to a system of equations
• 6 Classify a system of equations by graphing
• 7 Classify a system of equations
• 8 Solve a system of equations using substitution
• 9 Solve a system of equations using substitution: word problems
• 10 Solve a system of equations using elimination
• 11 Solve a system of equations using elimination: word problems
• 12 Solve a system of equations using augmented matrices
• 13 Solve a system of equations using augmented matrices: word problems
• 14 Solve a system of equations using any method
• 15 Solve a system of equations using any method: word problems

P. Two-variable linear inequalities

• 1 Does (x, y) satisfy the inequality?
• 2 Linear inequalities: solve for y
• 3 Graph a two-variable linear inequality
• 4 Write a linear inequality from a graph
• 5 Write two-variable inequalities: word problems
• 6 Is (x, y) a solution to the system of linear inequalities?
• 7 Solve systems of linear inequalities by graphing

Q. Piecewise-defined functions

• 1 Evaluate piecewise-defined functions
• 2 Graph piecewise-defined functions

R. Integer exponents

• 1 Powers with integer bases
• 2 Powers with decimal and fractional bases
• 3 Negative exponents
• 4 Multiplication rule for exponents
• 5 Division rule for exponents
• 6 Power rule for exponents
• 7 Simplify exponential expressions using the multiplication rule
• 8 Simplify exponential expressions using the division rule
• 9 Simplify exponential expressions using the multiplication and division rules
• 10 Simplify exponential expressions using the power rule
• 11 Simplify exponential expressions using exponent rules
• 12 Evaluate expressions using exponent rules
• 13 Identify equivalent exponential expressions I
• 14 Identify equivalent exponential expressions II

S. Rational exponents

• 1 Evaluate integers raised to positive rational exponents
• 2 Evaluate integers raised to rational exponents
• 3 Multiplication with rational exponents
• 4 Division with rational exponents
• 5 Power rule with rational exponents
• 6 Simplify expressions involving rational exponents

T. Scientific notation

• 1 Convert between standard and scientific notation
• 2 Compare numbers written in scientific notation
• 3 Add and subtract numbers written in scientific notation
• 4 Multiply numbers written in scientific notation
• 5 Divide numbers written in scientific notation

U. Sequences

• 1 Identify arithmetic and geometric sequences
• 2 Arithmetic sequences
• 3 Geometric sequences
• 4 Evaluate variable expressions for number sequences
• 5 Evaluate recursive formulas for sequences
• 6 Identify a sequence as explicit or recursive
• 7 Write variable expressions for arithmetic sequences
• 8 Write variable expressions for geometric sequences
• 9 Write a formula for a recursive sequence
• 10 Convert a recursive formula to an explicit formula
• 11 Convert an explicit formula to a recursive formula
• 12 Sequences: mixed review

V. Exponential functions

• 1 Evaluate an exponential function
• 2 Graph exponential functions
• 3 Match exponential functions and graphs I
• 4 Match exponential functions and graphs II
• 5 Domain and range of exponential functions: graphs
• 6 Domain and range of exponential functions: equations
• 7 Write exponential functions: word problems
• 8 Exponential growth and decay: word problems
• 9 Compound interest: word problems

W. Polynomials

• 1 Polynomial vocabulary
• 2 Model polynomials with algebra tiles
• 3 Add and subtract polynomials using algebra tiles
• 4 Add and subtract polynomials
• 5 Add polynomials to find perimeter
• 6 Multiply a polynomial by a monomial
• 7 Multiply two binomials using algebra tiles
• 8 Multiply two binomials
• 9 Multiply two binomials: special cases
• 10 Multiply polynomials using area models
• 11 Multiply polynomials
• 12 Multiply polynomials to find area
• 13 Divide polynomials by monomials
• 14 Divide polynomials using long division

X. Factor polynomials

• 1 GCF of monomials
• • New! LCM of monomials
• 2 Factor out a monomial
• 3 Factor quadratics using algebra tiles
• 4 Factor quadratics with leading coefficient 1
• 5 Factor quadratics with other leading coefficients
• 6 Factor quadratics: special cases
• 7 Factor by grouping
• 8 Factor using a quadratic pattern
• 9 Factor polynomials

Y. Quadratic functions

• 1 Characteristics of quadratic functions: graphs
• 2 Characteristics of quadratic functions: equations
• 3 Complete a function table: quadratic functions
• 4 Transformations of quadratic functions
• 5 Graph quadratic functions in vertex form
• 6 Write a quadratic function in vertex form
• 7 Graph quadratic functions in standard form
• 8 Match quadratic functions and graphs
• 9 Domain and range of quadratic functions: graphs
• 10 Domain and range of quadratic functions: equations
• 11 Write a quadratic function from its vertex and another point
• 12 Write a quadratic function from its x-intercepts and another point
• 13 Interpret parts of quadratic expressions: word problems

Z. Quadratic equations

• 1 Solve a quadratic equation using square roots
• 2 Solve a quadratic equation using the zero product property
• 3 Solve a quadratic equation by factoring
• 4 Complete the square
• 5 Solve a quadratic equation by completing the square
• 6 Solve a quadratic equation using the quadratic formula
• 7 Solve quadratic equations: word problems
• 8 Using the discriminant
• 9 Solve a system of linear and quadratic equations by graphing
• 10 Solve a system of linear and quadratic equations

AA. Linear, quadratic, and exponential functions

• 1 Identify linear and exponential functions from graphs
• 2 Identify linear, quadratic, and exponential functions from graphs
• 3 Identify linear and exponential functions from tables
• 4 Identify linear, quadratic, and exponential functions from tables
• 5 Write linear and exponential functions: word problems
• 6 Write linear and exponential functions from tables
• 7 Write linear, quadratic, and exponential functions from tables
• 8 Linear functions over unit intervals
• 9 Exponential functions over unit intervals
• 10 Describe linear and exponential growth and decay
• 11 Compare linear and exponential growth
• 12 Compare linear, exponential, and quadratic growth

BB. Absolute value functions

• 1 Complete a function table: absolute value functions
• 2 Graph an absolute value function
• 3 Domain and range of absolute value functions: graphs
• 4 Domain and range of absolute value functions: equations
• 5 Transformations of absolute value functions: translations and reflections
• 6 Transformations of absolute value functions: translations, reflections, and dilations

CC. Function operations

• 1 Add and subtract functions
• 2 Multiply functions
• 3 Find the inverse of a linear function

DD. Function transformations

• 1 Function transformation rules
• 2 Translations of functions
• 3 Reflections of functions
• 4 Dilations of functions
• 5 Transformations of functions
• 6 Describe function transformations

EE. Radical expressions

• 1 Simplify radical expressions
• 2 Simplify radical expressions with variables
• 3 Simplify radical expressions involving fractions
• 4 Multiply radical expressions
• 5 Add and subtract radical expressions
• 6 Simplify radical expressions using the distributive property
• 7 Simplify radical expressions using conjugates
• 8 Simplify radical expressions: mixed review
• 9 Convert between rational exponents and radicals

FF. Radical functions and equations

• 1 Evaluate square root functions
• 2 Domain and range of square root functions: graphs
• 3 Domain and range of square root functions: equations
• 4 Graph square root functions
• 5 Solve radical equations I
• 6 Solve radical equations II

GG. Rational functions and expressions

• 1 Rational functions: asymptotes and excluded values
• 2 Simplify rational expressions
• 3 Multiply and divide rational expressions
• 4 Add and subtract rational expressions
• 5 Simplify mixed rational expressions
• 6 Solve rational equations

HH. Data and graphs

• 1 Create line plots
• 2 Interpret line plots
• 3 Create histograms
• 4 Interpret histograms
• 5 Create bar graphs, line graphs, and histograms
• 6 Interpret bar graphs, line graphs, and histograms
• 7 Interpret circle graphs
• 8 Interpret stem-and-leaf plots
• 9 Box plots
• 10 Interpret graphics I
• 11 Interpret graphics II

II. Single-variable statistics

• 1 Identify biased samples
• 2 Mean, median, mode, and range
• 3 Calculate quartiles and interquartile range
• 4 Identify an outlier
• 5 Identify an outlier and describe the effect of removing it
• 6 Variance and standard deviation
• 7 Choose appropriate measures of center and variation

JJ. Bivariate statistics

• 1 Interpret a scatter plot
• 2 Outliers in scatter plots
• 3 Match correlation coefficients to scatter plots
• 4 Calculate correlation coefficients
• 5 Write equations for lines of best fit
• 6 Interpret lines of best fit: word problems
• 7 Find the equation of a regression line
• 8 Interpret regression lines
• 9 Analyze a regression line of a data set
• 10 Correlation and causation

KK. Probability

• 1 Theoretical probability
• 2 Experimental probability
• 3 Find probabilities using two-way frequency tables
• 4 Find conditional probabilities using two-way frequency tables
• 5 Outcomes of compound events
• 6 Identify independent and dependent events
• 7 Probability of independent and dependent events
• 8 Counting principle
• 9 Permutations
• 10 Permutation and combination notation

LL. Matrices

• 1 Matrix vocabulary
• 2 Matrix operation rules
• 3 Add and subtract matrices
• 4 Multiply a matrix by a scalar
• 5 Add and subtract scalar multiples of matrices
• 6 Multiply two matrices
• 7 Properties of matrices

Algebra 1 lessons

These lessons help you brush up on important math topics and prepare you to dive into skill practice!

Numbers and operations

• Adding and subtracting integers
• Multiplying and dividing integers
• Rational numbers
• Adding and subtracting rational numbers
• Multiplying and dividing rational numbers
• Square roots
• Irrational numbers
• Simple interest
• Percent change
• Transversals of parallel lines
• Interior angles of triangles
• Exterior angles of triangles
• The Pythagorean theorem
• The converse of the Pythagorean theorem
• Midpoint formula
• Distance formula

Number sequences

• Arithmetic sequences
• Geometric sequences

Expressions

• Writing algebraic expressions
• Evaluating expressions
• Simplifying expressions
• The distributive property
• Expanding expressions
• Factoring expressions

Single-variable equations

• Solving equations
• Multi-step equations
• Equations with infinitely many or no solutions

Single-variable inequalities

• Solving inequalities
• Compound inequalities

Absolute value equations and inequalities

• Absolute value equations
• Absolute value inequalities
• Relations and functions
• Independent and dependent variables
• Domain and range
• Inverse functions
• Proportional relationships
• Constant of proportionality
• Linear functions
• Linear equations
• Slope-intercept form
• Point-slope form
• Standard form of linear equations
• Equations of parallel and perpendicular lines
• Systems of equations
• Linear inequalities
• Systems of linear inequalities
• Properties of exponents
• Rational exponents
• Scientific notation
• Adding and subtracting numbers in scientific notation
• Multiplying and dividing numbers in scientific notation
• Exponential functions
• Exponential growth
• Exponential decay
• Quadratic equations
• Factoring quadratic equations
• Completing the square
• The quadratic formula
• Quadratic functions
• Absolute value functions
• Piecewise functions

Data and graphs

• Box and whisker plots
• Standard deviation and variance
• Scatter plots
• Correlation
• Line of best fit
• Two-way tables

Please ensure that your password is at least 8 characters and contains each of the following:

• a special character: @$#!%*?&

Free Printable Math Worksheets for Algebra 1

Created with infinite algebra 1, stop searching. create the worksheets you need with infinite algebra 1..

• Fast and easy to use
• Multiple-choice & free-response
• Never runs out of questions
• Multiple-version printing

Free 14-Day Trial

• Writing variable expressions
• Order of operations
• Evaluating expressions
• Number sets
• Adding rational numbers
• Adding and subtracting rational numbers
• Multiplying and dividing rational numbers
• The distributive property
• Combining like terms
• Percent of change
• One-step equations
• Two-step equations
• Multi-step equations
• Absolute value equations
• Solving proportions
• Percent problems
• Distance-rate-time word problems
• Mixture word problems
• Work word problems
• Literal Equations
• Graphing one-variable inequalities
• One-step inequalities
• Two-step inequalities
• Multi-step inequalities
• Compound inequalities
• Absolute value inequalities
• Discrete relations
• Continuous relations
• Evaluating and graphing functions
• Finding slope from a graph
• Finding slope from two points
• Finding slope from an equation
• Graphing lines using slope-intercept form
• Graphing lines using standard form
• Writing linear equations
• Graphing linear inequalities
• Graphing absolute value equations
• Direct variation
• Solving systems of equations by graphing
• Solving systems of equations by elimination
• Solving systems of equations by substitution
• Systems of equations word problems
• Graphing systems of inequalities
• Discrete exponential growth and decay word problems
• Exponential functions and graphs
• Writing numbers in scientific notation
• Operations with scientific notation
• Addition and subtraction with scientific notation
• Naming polynomials
• Adding and subtracting polynomials
• Multiplying polynomials
• Multiplying special case polynomials
• Factoring special case polynomials
• Factoring by grouping
• Dividing polynomials
• Graphing quadratic inequalities
• Completing the square
• By taking square roots
• By factoring
• With the quadratic formula
• By completing the square
• Simplifying radicals
• Adding and subtracting radical expressions
• Multiplying radicals
• Dividing radicals
• Using the distance formula
• Using the midpoint formula
• Simplifying rational expressions
• Finding excluded values / restricted values
• Multiplying rational expressions
• Dividing rational expressions
• Adding and subtracting rational expressions
• Finding trig. ratios
• Finding angles of triangles
• Finding side lengths of triangles
• Visualizing data
• Center and spread of data
• Scatter plots
• Using statistical models

• Solve equations and inequalities
• Simplify expressions
• Factor polynomials
• Graph equations and inequalities
• Advanced solvers
• All solvers
• Arithmetics
• Determinant
• Percentages
• Scientific Notation
• Inequalities

What can QuickMath do?

QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

• The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
• The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
• The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
• The calculus section will carry out differentiation as well as definite and indefinite integration.
• The matrices section contains commands for the arithmetic manipulation of matrices.
• The graphs section contains commands for plotting equations and inequalities.
• The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

Math Topics

More solvers.

• Add Fractions
• Simplify Fractions

Algebra 1 or elementary algebra includes the traditional topics studied in the modern elementary algebra course. Basic arithmetic operations comprise numbers along with mathematical operations such as +, -, x, ÷. While, algebra involves variables like x, y, z, and mathematical operations like addition , subtraction , multiplication , and division to form a meaningful mathematical expression.

Algebra helps in the representation of different situations or problems as mathematical expressions . The concepts that come under algebra 1 or elementary algebra include variables, evaluating expressions and equations, properties of equalities and inequalities, solving the algebraic equations and linear equations which have one or two variables, and so on.

What is Algebra 1?

Algebra 1  consists of the general/basic concepts of algebra. It introduces evaluating equations and inequalities, real numbers, and their properties, which include additive and multiplicative identities, inverse operations , and the distributive and commutative properties. In algebra 1, we will also be introduced to the concept of polynomials, and will also incorporate a bit of geometry to calculate the area, volume, and perimeters of shapes using algebraic expressions instead of numbers.

Algebra 1 or elementary algebra deals with solving the algebraic expressions for a viable answer. In algebra 1, simple variables like x, y, are represented in the form of an equation. Based on the degree of the variable the equations can be categorized into different types, namely linear equations, quadratic equations, cubic equations, and so on. Linear equations are of the forms of ax + b = c, ax + by + c = 0, ax + by + cz + d = 0. Elementary algebra based on the degree of the variables, branches out into quadratic equations and polynomials. A general form of representation of a quadratic equation is ax 2 + bx + c = 0, and for a polynomial equation, it is ax n + bx n-1 + cx n-2 + .....k = 0.

The rules for different properties under algebra 1 can be understood better as shown below,

Algebra 1 Topics

Algebra is divided into numerous topics to help for a detailed study. Algebra 1 is divided into 12 chapters and each chapter is divided into several lessons. These 12 chapters in Algebra 1 are given as:

Chapter 1: Real Numbers and Their Operations

Chapter 2: Linear Equations and Inequalities

• Variable expressions
• Linear Equations

Chapter 3: Introduction to Functions

• What Are Functions?
• Polynomial Functions
• Visualizing Functions Through Graphs
• Arithmetic and Geometric Progressions

Chapter 4: Graphing Lines

• Cartesian System
• Graphing linear equations
• Perpendicular Line
• Parallel Lines

Chapter 5: Solving Linear Systems

• Solving Linear Systems by Substitution
• Solving Linear Systems by Cross Multiplication
• System of Equations Solver
• Solutions of a Linear Equation

Chapter 6: Polynomials and Their Operations

• Polynomials
• Polynomial Expressions
• nth Degree Polynomial
• Multiplying Polynomials

Chapter 7: Factoring and Solving by Factorization

• What is a Factor?
• Factoring Methods
• Factorization of Algebraic Expressions
• Factorization of Quadratic Equations

Chapter 8: Exponents And Exponential Functions

• Exponential Functions
• Irrational Exponents
• Operations on Exponential Terms

Chapter 9: Rational Expressions and Equations

• Rational Numbers
• Rational Function
• Non-Integer Rational Exponents
• Simplifying Rational Expressions

Chapter 10: Radical Expressions and Equations

• Square and Square Root
• Rationalization
• Rationalize the Denominator

Chapter 11: Solving Quadratic Equations and Graphing Parabolas

• Square and Square Roots
• Quadratic Formula
• Graphing a Quadratic Function
• Complex Numbers and Complex Solutions

Chapter 12: Data Analysis And Probability

• Data Handling
• Probability and Statistics
• Categorical Data
• Permutations and Combinations

Laws of Algebra 1

The basic laws of algebra are the associative, commutative, and distributive laws that are presented in the table below:

Algebra 1 Formulas

Here are the list of formulas that are very useful in solving Algebra 1 problems.

• Algebraic identities :  (a + b) 2  = a 2  + 2ab + b 2 (a - b) 2  = a 2  - 2ab + b 2 (a + b)(a - b) = a 2  - b 2 (x + a)(x + b) = x 2  + x(a + b) + ab (a + b) 3  = a 3  + 3a 2 b + 3ab 2  + b 3 (a - b) 3  = a 3  - 3a 2 b + 3ab 2  - b 3 a 3  + b 3  = (a + b)(a 2  - ab + b 2 ) a 3  - b 3  = (a - b)(a 2  + ab + b 2 ) (a + b + c) 2  = a 2  + b 2  + c 2  + 2ab + 2bc + 2ca
• Properties of Exponents : a m . a n  = a m + n a m /a n  = a m - n (a m ) n  = an (ab) m  = a m . b m a 0  = 1 a -m  = 1/a m
• Linear Equations Formulas : General form: ax + by = c Slope Intercept Form : y = mx + b Two-Point Form :  y−y 1 =m(x−x 1 ) Intercept Form : x/a + y/b = 1 Vertical Line through (p, q): x = p Horizontal Line through (p, q): y = q
• Quadratic Equations Formulas: The standard form of quadratic equation  is ax 2  + bx + c = 0 Vertex form of quadratic equation is a (x - h) 2  + k = 0 Quadratic Formula:  The roots of a quadratic equation ax 2  + bx + c = 0 are given by x = [-b ± √(b² - 4ac)]/2a.
• Arithmetic Sequence Formulas : n th  term , a n =a 1 +(n−1)d Sum = n/2 [2a + (n - 1) d] (OR) n/2 [a 1 + a n ]
• Geometric Sequence Formulas : The  n th  term of the geometric sequence  is, a n  = a · r n - 1 . Sum of n terms , S n  = a (r n  - 1) / (r - 1) Sum of infinite terms , S = a / (1 - r)
• Average rate of change formula : [f(b) - f(a)] / (b - a)
• Compound Interest Formula : A = P (1 + r / n) n t  
• Statistics Formulas : Mean = (Sum of Observations) ÷ (Total Numbers of Observations) Mean of Grouped Data =  Σf i /N Median when 'n' is odd:  [(n + 1)/2] th  term; Median when 'n' is even: [(n/2) th  term + ((n/2) + 1) th  term]/2 Range  = Maximum - Minimum Interquartile Range = Upper quartile - Lower quartile

Difference Between Algebra 1 and Algebra 2

Algebra 1 and Algebra 2 can be distinguished based on the complexity and use of algebraic expressions. The following table explains the important differences between algebra 1 and algebra 2.

Tips and Tricks on Algebra 1

• To understand Algebra 1, we need to be familiar with the pre-algebra topics like integers, one-step equations, multistep equations , inequalities and equations, graphs and functions, percent, probabilities, an introduction to geometry, and, right triangles. Once we go through a refresher, then we can proceed to algebra 1.
• When multiplying two rational expressions in algebra, there is always a risk of getting false solutions or extraneous solutions so be careful with your calculations part.
• We can add polynomials by just adding the like terms to combine the two polynomials into one.

Important Notes on Algebra 1:

• The addition property of inequality: Adding the same number to each side of the inequality produces an equivalent inequality.
• Negative exponents : The reciprocals of the positive exponents in exponential functions.
• The quotient of powers property : It tells us that when we divide the powers with the same base we just have to subtract the exponents.
• The constants have a monomial degree of 0.

☛ Related Topics:

• Math Calculator
• Algebra Calculator
• Algebraic Formula Calculator

Algebra 1 Problems

Example 1: Using laws and properties of algebra 1, evaluate the expression (4 × (x + 2)), where x = 5.

Given, x = 5. Putting the value of x in 4 × (x + 2), we get, 4 × (5 + 2) = 4 × 7 = 28.

Example 2: Solve the given expression for the value of x, 4 + 3 = x. Solution:

Given, 4 + 3 = x. We will simply do the addition of the given expression and get the value of x. 4 + 3 is equal to 7, or x = 4 + 3 = 7.

Answer:  Therefore, the value of x is 7.

Example 3: Solve the given expression using rules of algebra 1 for the value of y, 2y + 16 = 32.

Given, 2y + 16 = 32. We will solve the equation for y.

2y = 32 - 16 gives 2y = 16, y = 8.

Answer:  Therefore, the value of y is 8.

go to slide go to slide go to slide

Book a Free Trial Class

Practice Questions on Algebra 1

Faqs on algebra 1, what will you learn in algebra 1.

Algebra 1  or Elementary algebra includes the basic traditional topics studied in the modern elementary algebra course. Basic arithmetic operations comprise numbers along with mathematical operations such as +, -, x, ÷. While, algebra involves variables as well like x, y, z, and mathematical operations like addition, subtraction, multiplication, and division to form a meaningful mathematical expression.

What is Considered Algebra 1?

Algebra 1 consists of the general concepts of algebra. It introduces evaluating equations and inequalities , real numbers, and their properties, which include additive and multiplicative identities , inverse operations, and the distributive and commutative properties.

What is the Difference Between Algebra 1 and Algebra 2?

The difference between Algebra 1 and Algebra 2 can be understood using the following points:

• Algebra 1 helps students to have the basic command in algebra topics, while algebra 2 increases complexity and understanding of the topics learned in algebra 1.
• In algebra 1, students learn how to manipulate exponents or polynomials and write them in simpler forms, etc, while in Algebra 2, students learn to apply the skills thus obtained in algebra 1 and also learn more difficult techniques.
• Algebra 1 is concentrated on solving equations and inequalities. But, algebra 2 concentrates on additional types of equations, such as exponential and logarithmic equations.
• Algebra 1 is essential to understand algebra 2, whereas, algebra 2 is essential for understanding concepts coming on calculus.

What is Standard Form in Algebra 1?

A standard form in Algebra 1 is a form of writing a given mathematical concept like an equation, number, or an expression in a form that follows certain rules.

How to Learn Algebra 1 Fast?

The concepts of algebra 1 can be mastered by following certain instructions. The key points given below will help you ensure a thorough graphing of elementary algebra.

• Focus on basic arithmetic concepts.
• Remember PEMDAS rule.
• Learn to distinguish clearly between the roles of variables , constants , exponents, and negative and positive numbers.
• Do a thorough revision of formulas.
• Work on practice problems.

What Grade is Algebra 1?

Algebra 1 or elementary algebra is the first math class you are required to take as part of your middle school. We study real numbers, exploring solving, writing, and graphing linear equations in this part of Algebra. Also, polynomials, as well as quadratic equations and functions are included in Algebra 1.

What Topics are Covered in Algebra 1?

The topics covered in algebra 1 are divided into different chapters. These chapters can be broadly classified into the following categories:

• Real Numbers and Their Operations
• Linear Equations and Inequalities
• An Introduction To Functions
• Graphing Lines
• Solving Linear Systems
• Polynomials and Their Operations
• Factoring and Solving by Factoring
• Exponents And Exponential Functions
• Rational Expressions and Equations
• Radical Expressions and Equations
• Solving Quadratic Equations and Graphing Parabolas
• Data Analysis And Probability

Is Algebra 1 or 2 Harder?

Algebra 1 is the building block of algebra 2. Algebra 2 is a higher and more complex course, hence algebra 2 is a lot harder than algebra 1.

What is Algebra 1 Equations?

The equations of algebra 1 include only linear equations and quadratic equations. Cubic equations and other higher-order equations are NOT a part of algebra 1.

What is the First Thing you Learn in Algebra 1?

The first thing students learn in algebra 1 is real numbers and their operations.

What are the Prerequisites to Understand Algebra 1 Better?

To understand Algebra 1, it is an advantage if you know the foundations of arithmetic, integers, fractions, decimals, percent, ratio, proportion, probabilities , an introduction to geometry, and, right triangles .

• Math Test Prep & More

Chapter 1: Introduction to Algebra

• Video 1: Definition of Terms
• Video 2: Negatives
• Video 3: Throwing Light on the Number One
• Video 4: Zero has an Effect

Chapter 2: Addition and Subtraction in Algebra

• Video 1: Addition and Subtraction in Algebra (FREE)

Chapter 3: Parenthesis

• Video 1: Parenthesis in Algebra

Chapter 4: Multiplication in Algebra

• Video 1: Multiplication in Algebra

Chapter 5: Exponent

• Video 1: Introduction and Laws of Exponents (FREE)
• Video 2: Exponents on negatives, Monomials and Polynomials

Chapter 6: Division in Algebra

• Video 1: Division in Algebra
• Video 2: Division of Polynomial by Polynomial

Chapter 7: Radicals

• Video 1: Introduction to Radicals
• Video 2: Operations on Radicals

Chapter 8: Factors and Multiples

• Video 1: Factors and Factorization (FREE)
• Video 2: Factoring Quadratics
• Video 3: Multiples

Chapter 9: Introduction to Fractions

• Video 1: Introduction to Algebra

Chapter 10: Operation on Fractions

• Video 1: Operation on Fractions

Chapter 11: Equations

• Video 1: Introduction to the Basic Concept of Equations
• Video 2: Transposing more than Once
• Video 3: Using the Reversal of the Order of Operation to Solve Equations
• Video 4: Solving Equations in which Variable is represent more than Once
• Video 5: Effect of Fraction on Solving Equations
• Video 6: Solving Simultaneous Equations by Elimination/Addition
• Video 7: Solving Simultaneous Equations by Substitution
• Video 8: Solving Quadratic Equations
• Video 9: Solving Quadratic Equations with the Quadratic Formula (FREE)

Chapter 12: Inequalities

• Video 1: Introduction to Inequalities
• Video 2: Graphing Inequalities

Chapter 13: Relation and Functions

• Video 1: Relation and Function

Basic Algebra Video for Beginners

SAMPLE ALGEBRA 1 VIDEOS

• Intro & Laws of Exponents
• Factors and Factorization (2)
• Addition and Subtraction in Algebra
• Solving Equations Using the Quadratic Fomula

Get 50% OFF Premium Course

50% Off all Courses Dismiss

You can excel with Caddell!

Give us a call (917) 722-0677

Algebra 1 Help | Free Algebra 1 Lessons & Practice Problems

Go through the lessons and practice problems below to help you learn Algebra 1 and excel in school. We’ll track your progress and help you identify your strengths and weaknesses. Our Algebra 1 lessons is available to everyone, but you need to create an account in order to access the practice questions and track your progress.

Algebra 1: 01-Introduction to Algebra

Algebra 1: 02-solving equations, algebra 1: 03-solving inequalities, algebra 1: 04-functions, algebra 1: 05-linear functions, algebra 1: 06-systems of equations and inequalities, algebra 1: 07-exponents and exponential functions, algebra 1: 08-polynomials & factoring, algebra 1: 09-quadratic functions and equations, algebra 1: 10-radical expressions & functions, protected: order – march 16, 2020 @ 08:13 pm, algebra 1: 11-rational expressions & functions, algebra 1: 12-data analysis & probability, i. introduction to algebra, lesson: verbal to algebraic expressions & equations.

• Practice 01
• Practice 02
• Practice 03
• Practice 04

Lesson: Declaring Variables

Example: Mike is 3 inches taller than Semande. If Semande is s inches tall, express Mike’s height in terms of s .

Lesson: Order of Operations

Lesson: Real Numbers (Including Rational vs. Irrational Numbers)

Lesson: order of operations with substitution.

Lesson: Adding and Subtracting Real Numbers

Lesson: Multiplying and Dividing Real Numbers

Lesson: Properties of Real Numbers

Lesson: Distributive Property

Lesson: Finding the Solution to an Equation

II. Solving Equations

Lesson: solving algebraic equations (one-step).

Lesson: Solving Multi-Step Equations

Lesson: Multi-Step Equations (Variable on Both Sides)

Lesson: Linear Equations

Lesson: Solving Proportions

Lesson: Solving Complex Proportions

Lesson: Percents

Example: What is 48% of 30?

Lesson: Percent Change

Example: What is the percent change from 50 to 72?

III. Solving Inequalities

Lesson: graphing inequalities.

Lesson: Solving Inequalities Using Addition or Subtraction

Lesson: Solving Inequalities Using Multiplication or Division

Lesson: Solving Inequalities

Lesson: Solving Compound Inequalities

Lesson: Solving and Graphing Compound Inequalities

Lesson: Absolute Value Equations

Lesson: Absolute Value Inequalities

IV. Functions

Lesson: using graphs to relate two quantities.

Lesson: Patterns and Linear Functions

Lesson: Patterns and Non-Linear Functions

Example: For the set of ordered pairs, write the rule that represents the function. (0,1),(1,3),(2,9),(3,27),(4,81)

Lesson: Graphing a Function Rule

Lesson: Writing a Function Rule

Lesson: Formalizing Relations and Functions

Example: Is the following relation shown a function? (-2, 0.5), (0, 2.5), (4, 6.5), (5, 2.5)

V. Linear Functions

Lesson: slope.

• Practice 01 – Slope from a graph & from 2 points
• Practice 02 – Slope from a graph & from 2 points
• Practice 03 – Slope from 2 Points
• Practice 04 – Slope from 2 Points

Lesson: Equation of a Line

Lesson: point slope form, lesson: finding the equation of a line, given the slope and a point, lesson: standard form of the equations of a line.

Lesson: Finding the Equation of a Line, Given Two Points

Lesson: equations of parallel and perpendicular lines.

Lesson: Scatterplots and Trendlines

• If the trend continues, what is the expected y-value when x is 8, based on the scatterplot shown below?

Lesson: Graphing Absolute Value Functions

VI. Systems of Equations and Inequalities

Lesson: solving a system of equations graphically.

Lesson: Solving a System of Equations Using Substitution

Lesson: Solving a System of Equations Using Elimination

Lesson: Applications of Linear Systems

Lesson: word problems – system of equations, lesson: graphing linear inequalities.

• What system of inequalities is shown below?

Lesson: Solving a System of Linear Inequalities

VII. Exponents and Exponential Functions

Lesson: positive exponents.

Lesson: Negative Exponents

Lesson: Multiplying & Dividing Terms that have Exponents

Lesson: Graphs of Exponential Functions

Lesson: Exponential Growth

Lesson: exponential decay.

VIII. Polynomials and Factoring

Lesson: adding & subtracting polynomials.

Lesson: Multiplying a Polynomial by a Monomial

Lesson: Dividing a Polynomial by a Monomial

Lesson: Factor Out the Greatest Common Factor

Lesson: Multiplying Binomials

Lesson: Multiplying a Binomial by a Trinomial

Lesson: Multiplying a Trinomial by a Binomial

Lesson: Multiplying Binomials (Special Cases)

Lesson: Factoring Quadratics

Lesson: Difference of Perfect Squares

Lesson: Factoring by Grouping

IX. Quadratic Functions and Equations

Lesson: evaluating quadratic functions.

Lesson: Properties of Quadratic Functions

Lesson: Solving Quadratic Equations by Taking Square Roots

Lesson: Solving Equations by Factoring

Lesson: Completing the Square

Lesson: Solve Equations by Completing the Square

Lesson: Vertex Form of a Quadratic Function by Completing the Square

Lesson: Solving a System of Equations Graphically (Linear & Quadratic)

Lesson: Solving a System of Equations Using Substitution (Linear & Quadratic)

X. Radical Expressions and Functions

Lesson: pythagorean theorem, lesson: simplifying radical expressions.

Lesson: Adding and Subtracting Radical Expressions

Lesson: Multiplying Radical Expressions

Lesson: Dividing Radical Expressions

Lesson: Solving Equations That Have Radical Terms

Lesson: Graphing Square Root Functions

Lesson: SOHCAHTOA (sine, cosine, tangent)

Lesson: Inverse Trigonometric Functions

XI. Rational Expressions and Functions

Lesson: simplifying rational expressions.

Lesson: Multiplying and Dividing Rational Expressions

Lesson: Dividing Polynomials

Lesson: Adding and Subtracting Rational Expressions

Lesson: Fractional Equations

Lesson: Solving Rational Equations

Lesson: Inverse Variation

Lesson: Graphing Rational Functions

XII. Data Analysis and Probability

Lesson: matrices.

• What are the dimensions of this matrix?

Lesson: Frequency, Cumulative Frequency, & Histograms

Lesson: measures of central tendency & dispersion, lesson: box & whisker plot, lesson: samples and surveys, lesson: permutations and combinations.

Lesson: Theoretical vs. Experimental Probability

Lesson: probability of compound events.

Sign up for a free membership on our algebra 1 lessons & start learning today. Learn about the importance of learning algebra here .

Online Test Prep for Schools

Terms of Service

Privacy Policy

TACHS Vocabulary  

Past Regents Exams

As seen on:

Algebra 1 Topics and Concepts

• The Albert Team
• Last Updated On: March 1, 2022

Looking for all the topics in Algebra 1? 

Algebra 1 is a math course exploring how to use variables, numbers, and mathematical operations to solve problems.  Explore below for a list of all Algebra 1 topics, a summary of the typical Algebra 1 course, and a discussion of what makes  Algebra 1 unique.

What We Review

What topics are in Algebra 1?

The below sequence of themes, concepts, and skills covers all of the most commonly taught Algebra 1 topics. As a bonus, many of the topics include links directly to Albert’s  Algebra 1 practice problems .

• Parts of an Expression
• Combining Like Terms
• Simplify and Evaluate Expressions
• Equivalent Expressions
• Algebraic Fractions
• Complex Fractions
• Exponent Laws and Notation
• Simplify Roots
• Radical Laws and Notation
• Rational Exponents
• Rationalize Denominators
• Polynomial Operations
• Interpret Expression Structure
• Factorization Procedures
• Factor Polynomials
• Units and Quantitative Reasoning
• One-Step Equations
• Two-Step Equations
• Multi-Step Equations
• The Coordinate Plane
• Understand Slope
• Slope-Intercept Form
• Point-Slope Form
• Standard Form
• Transformations of Linear Functions
• Parallel Lines
• Perpendicular Lines
• Understand Inequalities
• One-Step Inequalities
• Two-Step Inequalities
• Multi-Step Inequalities
• Compound Inequalities
• Linear Inequalities: Solve and Graph
• Linear Inequalities: Write and Apply
• Understand Systems of Equations
• Solve Systems of Equations
• Systems of Inequalities
• Understand Functions
• Function Notation
• Interpret and Model Functions
• Operations on Functions
• Composite Functions
• Inverse Functions
• Arithmetic Sequences
• Geometric Sequences
• Mixed Sequences
• Recursive Formulas for Sequences
• Exponential Growth and Decay
• Exponential Functions
• Piecewise Functions
• Absolute Value and Step Functions
• Function Transformations
• Explore Quadratic Functions
• Factored Form
• Vertex Form
• Understand Solving Quadratics or Polynomials
• Solve Quadratics: Factoring
• Solve Quadratics: Complete the Square
• Solve Quadratics: Graphing
• Solve Quadratics: Quadratic Formula
• Irrational Numbers
• Represent Data
• One-Variable Statistics
• Standard Deviation
• Two-Way Tables
• Scatter Plots

What is Algebra 1?

Algebra 1 is a high school math course exploring how to use letters (called variables) and numbers with mathematical symbols to solve problems.

Algebra 1 typically includes evaluating expressions, writing equations, graphing functions, solving quadratics, and understanding inequalities. Algebra 1 focuses on a variety of different types of functions such as linear, quadratic, and cubic — all of which are represented in traditional “function notation” as well as graphed on the coordinate plane.

Students in Algebra 1 typically build conceptual understanding and gain procedural fluency of concepts that are required for more advanced math subjects such as Geometry and Calculus. Abstracting real-life situations into mathematical models (expressions, equations, functions) is a key part of success in Algebra 1.

What grade is Algebra 1?

Algebra 1 is typically taught late in middle school or early in high school .

In the United States, 9th grade (freshman year) seems to be the most common grade for students to take an Algebra 1 class. Some high schools also offer Algebra 1 to 10th graders.

On the other end of the spectrum, many middle schools offer students an Algebra 1 course as early as 8th grade or even 7th grade for more advanced math students.

Is Algebra 1 hard?

It’s impossible to say exactly how easy or difficult Algebra 1 for each individual student. Students who have a strong background in middle school math topics (fractions, decimals, ratios, percents, negative numbers) should find an Algebra 1 course relatively accessible.

However, for many students, Algebra 1 will be quite a difficult challenge.

In Algebra 1, there are dozens of quickly-moving topics and skills that build on each other as the curriculum progresses. Having strong arithmetic skills is an incredibly important prerequisite for gaining confidence in an Algebra 1 course. If a student starts to get slightly behind on a specific concept, it’s likely those misconceptions will build into further confusion rather quickly.

If you’re looking for student practice at a wide range of difficulty levels, consider trying Albert’s Algebra 1 practice questions.

What is the difference between Algebra 1 and 2?

As the names suggest, there are many core similarities between Algebra 1 and Algebra 2. Both courses work extensively with variables and functions as well as using mathematical operations to find unknown quantities.

Students in Algebra 1 are typically expected to gain mastery of a wide variety of techniques for solving linear and quadratic equations. Algebra 1 focuses on “doing the same thing to both sides of the equation” to solve equations. With the lens of linear and quadratic functions, Algebra 1 pushes students to find roots (solutions) leveraging multiple methods (including the quadratic formula ), graph function relationships on the coordinate plane, and convert among various forms of quadratics .

Algebra 2 is an advanced expansion of the ideas from Algebra 1. Students in Algebra 2 are introduced to a much wider range of functions including logarithms, radicals, and rational functions. Trigonometric functions often become an important new focus area during Algebra 2 as students begin exploring the unit circle. Algebra 2 typically includes solutions with real and complex numbers — including arithmetic with imaginary numbers — as well as conic sections such as hyperbolas, parabolas, and ellipses. Basic parabolas that students used to think had no solutions during Algebra 1 are actually revealed to have two “imaginary” solutions in Algebra 2.

In general, Algebra 2 includes a wider and more intricate variety of function types than those covered in Algebra 1 topics. 

Interested in a school license?​

Popular posts.

AP® Score Calculators

Simulate how different MCQ and FRQ scores translate into AP® scores

AP® Review Guides

The ultimate review guides for AP® subjects to help you plan and structure your prep.

Core Subject Review Guides

Review the most important topics in Physics and Algebra 1 .

SAT® Score Calculator

See how scores on each section impacts your overall SAT® score

ACT® Score Calculator

See how scores on each section impacts your overall ACT® score

Grammar Review Hub

Comprehensive review of grammar skills

AP® Posters

Download updated posters summarizing the main topics and structure for each AP® exam.

Free Math Printable Worksheets with Answer Keys and Activities

Other free resources.

Feel free to download and enjoy these free worksheets on functions and relations. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Long Division with Remainders
• Long Division with Remainders #2 (Zeros in the Quotient)
• Long Division with 2 Digit Divisors
• Whole Number by Unit Fraction
• Equation of Circle
• Simplify Imaginary Numbers
• Adding and Subtracting Complex Numbers
• Multiplying Complex Numbers
• Dividing Complex Numbers
• Dividing Complex Number (Advanced)
• End of Unit, Review Sheet
• Distance Formula
• Simplify Rational Exponents (Algebra 2)
• Solve Equations with Rational Exponents (Algebra 2)
• Solve Equations with variables in Exponents (Algebra 2)
• Exponential Growth (no answer key on this one, sorry)
• Compound Interest Worksheet #1 (No logs)
• Compound Interest Worksheet (Logarithms required)
• Factor Trinomials Worksheet
• Factor by Grouping
• Domain and Range (Algebra 1)
• Functions vs Relations (Distinguish function from relation, state domain etc..) (Algebra 2)
• Evaluating Functions (Algebra 2)
• 1 to 1 Functions (Algebra 2)
• Composition of Functions (Algebra 2)
• Inverse Functions Worksheet (Algebra 2)
• Operations with Functions (Algebra 2)
• Functions Review Worksheet (Algebra 2)
• Logarithmic Equations
• Properties of Logarithms Worksheet
• Product Rule of Logarithms
• Power Rule of Logarithms
• Quotient Rule of Logarithms
• Solve Quadratic Equations by Factoring
• Quadratic Formula Worksheets (3 different sheets)
• Quadratic Formula Worksheet (Real solutions)
• Quadratic Formula (Complex solutions)
• Quadratic Formula (Both real and complex solutions)
• Discriminant and Nature of the Roots
• Solve Quadratic Equations by Completing the Square
• Sum and Product of Roots
• Radical Equations
• Mixed Problems on Writing Equations of Lines
• Slope Intercept Form Worksheet
• Standard Form Worksheet
• Point Slope Worksheet
• Write Equation of Line from the Slope and 1 Point
• Write Equation of Line From Two Points
• Equation of Line Parallel to Another Line and Through a Point
• Equation of Line Perpendicular to Another Line and Through a Point
• Slope of a Line
• Perpendicular Bisector of Segment
• Write Equation of Line Mixed Review
• Word Problems
• Multiplying Monomials Worksheet
• Multiplying and Dividing Monomials Sheet
• Adding and Subtracting Polynomials worksheet
• Multiplying Monomials with Polynomials Worksheet
• Multiplying Binomials Worksheet
• Multiplying Polynomials
• Simplifying Polynomials
• Factoring Trinomials
• Operations with Polynomials Worksheet
• Dividing Radicals
• Simplify Radicals Worksheet
• Adding Radicals
• Multiplying Radicals Worksheet
• Radicals Review (Mixed review worksheet on radicals and square roots)
• Rationalizing the Denominator (Algebra 2)
• Radical Equations (Algebra 2)
• Solve Systems of Equations Graphically
• Solve Systems of Equations by Elimination
• Solve by Substitution
• Solve Systems of Equations (Mixed Review)
• Activity on Systems of Equations (Create an advertisement for your favorite method to Solve Systems of Equations )
• Real World Connections (Compare cell phone plans)
• Identifying Fractions

Trigonomnetry

• Law of Sines and Cosines Worksheet (This sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well as on using both formulas to solve for a single triangle's side or angle)
• Law of Sines
• Ambiguous Case of the Law of Sines
• Law of Cosines
• Vector Worksheet
• Sine, Cosine, Tangent, to Find Side Length
• Sine, Cosine, Tangent Chart
• Inverse Trig Functions
• Real World Applications of SOHCATOA
• Mixed Review
• Unit Circle Worksheet
• Graphing Sine and Cosine Worksheet
• Sine Cosine Graphs with Vertical Translations
• Sine, Cosine, Tangent Graphs with Phase Shifts
• Sine, Cosine, Tangent Graphs with Change in Period, Amplitude and Phase Shifts (All Translations)
• Tangent Equation, Graph Worksheet
• Graphing Sine, Cosine, Tangent with Change in Period
• Cumulative, Summative Worksheet on Periodic Trig Functions - period, amplitude, phase shift, radians, degrees,unit circle
• Ratio and Proportion
• Similar Polygons
• Area of Triangle
• Interior Angles of Polygons
• Exterior Angles of Polygons

• Identifying Fractions Worksheet
• Associated Powerpoint
• Simplify Fractions Worksheet (Regular Difficulty)
• Associated PowerPoint
• Simplify Fractions Worksheet (Challenging Difficulty level for advanced learners)
• System of Linear Equations Worksheet
• System of Linear Equations - Real World Application
• Compositions of Reflections. Reflections Over Intersecting Lines as Rotations

All of these worksheets and activities are available for free so long as they are used solely for educational, noncommercial purposes and are not distributed outside of a specific teacher's classroom.

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

Popular pages @ mathwarehouse.com.

Need more information about an AoPS Academy course?

Ask AoPS Academy!

Academic Year Mathematics

In Algebra 1, students engage with various expressions from an algebraic and geometric perspective. They learn to solve linear and quadratic equations as well as how to represent expressions in the Cartesian plane. Students are imbued with a deep understanding of the general concept behind functions and are introduced to several important types of functions. Students are also introduced to the rich field of complex numbers. As with previous courses, we continue to emphasize challenging word problems to train students how to apply their new tools. Advanced problems from contests such as MATHCOUNTS, AMC 8, and AMC 10 are included throughout the course.

Homework and exams

Students should expect to spend 1-3 hours on homework every week. Homework will be assigned and completed through the class website, where students receive immediate feedback on their work. Students will take three in-class exams throughout the course.

The textbook for the course is Introduction to Algebra by Art of Problem Solving.

Hardcopy and lifetime digital access to the online version of the textbook are included in the tuition and fees of this course.

Sample Problems

Below are examples of some of the types of problems that students will encounter in our Algebra 1 course.

Virtual Campus Schedule

Fremont, CA Campus Schedule

Irvine, CA Campus Schedule

Mountain View, CA Campus Schedule

Pleasanton, CA Campus Schedule

San Diego, CA Campus Schedule

Santa Clara, CA Campus Schedule

Lexington, MA Campus Schedule

Gaithersburg, MD Campus Schedule

Morrisville, NC Campus Schedule

Princeton, NJ Campus Schedule

Millburn, NJ Campus Schedule

Frisco, TX Campus Schedule

Vienna, VA Campus Schedule

Bellevue, WA Campus Schedule

Redmond, WA Campus Schedule

Virtual Campus

Massachusetts, north carolina.

Our family regularly talks about AoPS. We try to think about how our life would be different without you all. What if my daughter hadn't learned to love math? What if she never experienced being pushed to her limits? Overcoming failure? She wouldn't be who she is. And she is AWESOME! A very proud mom here. We are so fortunate that we found AoPS Academy. I credit AoPS for much of her confidence.

AoPS Academy parent to 16-year-old engineering major at UW and future cancer researcher

Instructor To Be Announced

We have not yet announced an instructor for this class, but we are working on it! It can take time to match all of our classes to our instructors' daily schedules. However, all AoPS instructors are exceptional educators, and each section of this course will cover the same basic material. Students can confidently enroll in this class and expect that one of our outstanding instructors will be assigned soon.

Ask AoPS Academy

I have a question about:

Your information will be used in accordance with our Privacy Policy .

• Skip to primary navigation
• Skip to main content
• Skip to primary sidebar
• Skip to footer

Demme Learning

Building Lifelong Learners

• Default Sort Order
• Alphabetical: A to Z

Customer Service: M-Th 8:30am - 6pm ET Live Chat • 888-854-6284 • Email

Algebra 1 Prep: Understanding Readiness, Foundational Concepts, and What to Do if Your Student Needs a Boost

Demme Learning · August 14, 2024 · Leave a Comment

Have you ever heard someone say, “They threw me in the deep end and told me to swim!”? While some students may find diving into new topics exhilarating, Algebra 1 demands a solid foundation to ensure success, and students approaching this level need to be well-prepared for their next mathematics deep dive.

Ensuring your student is ready for Algebra 1 is crucial. Pre-Algebra lays this groundwork, teaching essential skills like understanding math language and manipulating fractions. Beyond mathematical knowledge, Pre-Algebra fosters the patience and problem-solving techniques needed to perform well in Algebra 1 in the future, as well as the perseverance and focus required.

They may have completed their Pre-Algebra coursework, but how do you know if your student has the algebra readiness they need to move on to Algebra 1?

Before continuing to the next steps in your Algebra journey on autopilot, take some time to ensure your student is ready for more complex material. Here are some things to know as you prepare for the future.

What is Readiness?

It’s easy to assume that students are ready for the next subject when they’ve completed a level of curricula or are moving on to the next school year, but there’s more to readiness than turning the page on the next chapter of your mathematics textbook. Readiness is the measure of subject knowledge, behavioral patterns, and work habits that ensure a student is prepared to tackle the next level of learning.

Foundational Concepts for Algebra 1

There are many prerequisite skills needed to prepare for an Algebra 1 course. Here are just a few of the key concepts that need to be mastered:

Write a fraction in its simplest form, write mixed numbers as improper fractions, and perform all operations on fractions.

Order of Operations

Simplify using order of operations, including expressions containing absolute value and exponents.

Solve Single Variable Equations

Use the solution to an equation to evaluate an expression, write a one-variable equation from a word problem and solve it, and solve an equation that has variables on both sides.

Inequalities

Solve one- and two-step inequalities and graph single-variable inequalities on a number line.

Reference a formula sheet to find an unknown formula, apply formulas for area, perimeter, & volume, and label a solution with the correct units.

Number Relationships

Name all factors of a number, and use proportions to solve problems.

Coordinate Plane

Name each quadrant and axis of the coordinate plane, plot ordered pairs in any quadrant of the coordinate plane, and determine the horizontal or vertical distance between ordered pairs (points) on the coordinate plane.

A complete list of foundational skills can be found here . These concepts are like tools in a toolbox. In Algebra 1, students will learn to implement these tools to glean relevant information from word problems and when solving more complex equations.

Determining Your Student’s Skill Level

Once you are aware of the prerequisite concepts, the next step is assessing your student’s skill level. This can be done through a readiness check. A readiness check is not a test but rather an evaluative assessment that can help determine where your student is at in their algebra journey.

Demme Learning provides an Algebra 1 Readiness Check, which can be found here and in our Digital Toolbox . It is highly recommended that your student take this assessment before beginning Algebra 1.

If Your Student Needs a Boost

Every student can use extra support from time to time. If you’ve completed a Readiness Check and found gaps in your student’s understanding, here are a few things that may help.

Encourage a Growth Mindset

Your student can develop a solid working knowledge of Algebra. Help them feel confident in their ability to grow by reminding them that every concept can be acquired with enough focus and practice.

Take Time to Review

Extra practice can make all the difference when building math confidence. Don’t be afraid to spend some extra time reviewing troublesome skills before moving on to the next level.

Take Advantage of Demme Learning’s Bridge Materials

This free set of skills lessons has been created to aid in preparing students for Algebra. They cover some of the essential concepts and skills needed for success in an Algebra course that may have been forgotten or not covered in great depth in a student’s previous studies. You can find our Bridge Materials in the drop-down menu bar under Products once you log in to your digital Demme Learning Account. Here, you’ll find access to the Readiness Check assessment and other individually focused videos and concept lessons listed below. Concepts like the Order of Operations, Proportions, The Coordinate Plane, and many other crucial skills to Algebra 1 are reviewed and retaught.

Whether your student feels ready for the challenge of Algebra 1 or is a little unsure, there’s still time to lay a solid foundation for their future. With the right tools and preparation, your student can take on the next challenges with confidence.

Our Bridge Materials are available in the Digital Toolbox. Click here to register for your free account !

Reader Interactions

Leave a reply cancel reply.

Your email address will not be published. Required fields are marked *

Save my name, email, and website in this browser for the next time I comment.

Related Stories

The Stories Heard on the Road: My Summer Adventures with Demme Learning

Math in Soccer: Analyzing Angles, Trajectories, and Goals

Mathematics in Sports: The Numbers Behind the Games

Game Central

• AI Math Solver Graphing Calculator Popular Problems Worksheets Study Guides Cheat Sheets Calculators Verify Solution
• Solutions Integral Calculator Derivative Calculator Algebra Calculator Matrix Calculator More...
• Graphing Line Graph Exponential Graph Quadratic Graph Sine Graph More...
• Calculators BMI Calculator Compound Interest Calculator Percentage Calculator Acceleration Calculator More...
• Geometry Pythagorean Theorem Calculator Circle Area Calculator Isosceles Triangle Calculator Triangles Calculator More...
• Tools Notebook Groups Cheat Sheets Worksheets Study Guides Practice Verify Solution

• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Expanded Form Mean, Median & Mode
• Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Prove That Logical Sets Word Problems
• Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Coterminal Angle Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
• Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
• Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
• Linear Algebra Matrices Vectors
• Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
• Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
• Physics Mechanics
• Chemistry Chemical Reactions Chemical Properties
• Finance Simple Interest Compound Interest Present Value Future Value
• Economics Point of Diminishing Return
• Conversions Roman Numerals Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time Volume
• Pre Algebra
• Pre Calculus
• Linear Algebra
• Trigonometry
• Conversions

Number Line

• x^{2}-x-6=0
• -x+3\gt 2x+1
• line\:(1,\:2),\:(3,\:1)
• prove\:\tan^2(x)-\sin^2(x)=\tan^2(x)\sin^2(x)
• \frac{d}{dx}(\frac{3x+9}{2-x})
• (\sin^2(\theta))'
• \lim _{x\to 0}(x\ln (x))
• \int e^x\cos (x)dx
• \int_{0}^{\pi}\sin(x)dx
• \sum_{n=0}^{\infty}\frac{3}{2^n}
• Is there a step by step calculator for math?
• Symbolab is the best step by step calculator for a wide range of math problems, from basic arithmetic to advanced calculus and linear algebra. It shows you the solution, graph, detailed steps and explanations for each problem.
• Is there a step by step calculator for physics?
• Symbolab is the best step by step calculator for a wide range of physics problems, including mechanics, electricity and magnetism, and thermodynamics. It shows you the steps and explanations for each problem, so you can learn as you go.
• How to solve math problems step-by-step?
• To solve math problems step-by-step start by reading the problem carefully and understand what you are being asked to find. Next, identify the relevant information, define the variables, and plan a strategy for solving the problem.
• Practice, practice, practice Math can be an intimidating subject. Each new topic we learn has symbols and problems we have never seen. The unknowing...

We want your feedback

Please add a message.

Message received. Thanks for the feedback.

Skip to Content

Current Students

Interested in more? Search Courses

• Search Input Submit Search

DU Professor Helps Solve Famous 70-Year-Old Math Problem

Jordyn reiland.

[email protected]

Assistant Professor Mandi Schaeffer Fry is the first faculty member to be published in the Annals of Mathematics since the 1880s.

University of Kaiserslautern Professor Gunter Malle, University of Denver Assistant Professor Mandi Schaeffer Fry and University of Valencia Professor Gabriel Navarro pose for a photo after announcing their theorem in Oberwolfach, Germany.

Whether it be flying trapeze, participating in competitive weightlifting or solving math problems that have confounded academics for decades, Mandi Schaeffer Fry enjoys chasing the next adventure.

Schaeffer Fry, who joined the University of Denver’s Department of Mathematics in the fall of 2023, will be the first faculty member since the 1880s to be published in the Annals of Mathematics , widely seen as the industry’s most prestigious journal.

In 2022, Schaeffer Fry helped complete a problem that dates to 1955—mathematician Richard Brauer’s Height Zero Conjecture.

“Maybe one of the most challenging parts, other than the math itself, was the knowledge of the weight that this would have on the field,” Schaeffer Fry says. “If you’re going to make an announcement like this, you have to be darn sure that it’s absolutely correct.”

Over the years, number crunchers have worked on the problem at universities across the globe, and some found partial solutions; however, the problem was not completed until now.

“Mandi’s accomplishment is exciting. Solving Brauer's Height Zero Conjecture is remarkable,” Mathematics Department Chair Alvaro Arias says. 

The work is also a testament to DU’s achievement as a Research 1 (R1) institution.

Fry and her collaborators—University of Kaiserslautern Professor Gunter Malle, University of Valencia Professor Gabriel Navarro and Rutgers University Professor Pham Huu Tiep—worked around the clock over the course of three months in eight-hour shifts during the summer of 2022 to find a solution.

In April, that work was accepted for publication in the Annals of Mathematics.

'Brauer's Height Zero Conjecture (BHZ) was the first conjecture leading to the part of my field studying 'local-global' problems in the representation theory of finite groups, which seek to relate properties of groups with properties of certain nice smaller subgroups, letting us 'zoom in' on the group using just a specific prime number and simplify things," Schaeffer Fry says. 

"The BHZ gives us a way to tell from the character table of a group (a table of data that encodes lots, but not all, information about the group) whether or not certain of these subgroups, called defect groups, have the commutativity property," she adds.

This paper was especially meaningful to Schaeffer Fry as she had always wanted to work with Malle, Tiep and Navarro as they have been her primary mentors. Tiep was her PhD advisor and this was the first time they had worked together since then.

Fry believes she has solidified her place in the field and knows she’ll likely never top this accomplishment, but she’s always looking for the next adventure—whether that’s in or out of the classroom.

Flying high and pumping iron

When Schaeffer Fry isn’t on DU’s campus working with students or conducting research, you can find her flying trapeze and competitive weightlifting.

Schaeffer Fry became involved in competitive weightlifting during graduate school, and, in the last year of her PhD at the University of Arizona, she defended her dissertation one day and got on a plane and competed at the national level for “university-aged” athletes—which included Olympians.

While she now lifts weights more casually, Schaeffer Fry competed last September in an over-35 competition and qualified for the USA Weightlifting Masters National Championships.

It was a “field trip” during a conference in Berkeley, California, in 2018 that led Fry to become enamored with flying trapeze.

In fact, she enjoyed it so much she signed up to be a member of Imperial Flyers, an amateur flying trapeze cooperative located in Westminster. Once she found out about the sport, her previous experience as a gymnast made it a natural fit.

Not only is she working on her own intermediate tricks, she’s also a “teaching assistant” at Fly Mile High, the state’s only flying trapeze and aerial fitness school.

“It’s exhilarating; it’s gotten me a bit over my fear of heights,” she says.

Related Articles

A Geology Expert’s Guide to Colorado’s Coolest Rock Formations

Research to Watch

Storytelling, Shaving Cream and Slime: Why Students Love Professor Nancy Sasaki

AI Math Solver Question Helper 4+

Ai tutor to help math question, damjibhai chotaliya, designed for iphone.

• Offers In-App Purchases

iPhone Screenshots

Description.

Using the latest AI, AI Math Solver Question Helper helps streamline your studies with tools for scan, writing, problem solving, help to homework and more. Simply click photograph of your math problem, and AI Math Solver Question Helper will review it, offering in depth insights and step-by-step breakdowns. This approach to understanding solutions and enhances your math abilities. AI Math Solver Question Helper is an advanced tool designed to assist students at all learning levels. By simply taking a photo of a math problem, users get instant, comprehensive solutions with detailed step by step explanations. The app covers a wide range of topics, from basic arithmetic to complex calculus, transforming challenging problems into manageable tasks. Its user-friendly design and powerful AI technology promote a deeper understanding of mathematical concepts and enhance problem solving skills. Ideal for homework help, exam preparation, or mastering difficult concepts, AI Math Solver Question Helper makes learning math both efficient and engaging. FEATURES INCLUDED: - Scan to Solve with AI Problem Solver - Get quick Word problem & Geometry solution - Math, Science, & Liberal Arts any subject can explained - Step by Step Explanations for any Query - Customized AI Tutor COVERED MATH TOPICS: - Solve Math problems for elementary to college and beyond every level - Algebra 1 & 2: covering linear, quadratic, exponential equations, and more - Calculus includes limits, differential equations, integrals, and more - Trigonometry: angles, functions, identities, and more - Simplify, factor, and evaluate polynomials and fractions - Matrix operations: multiplication, arithmetic, and more - Probability and statistics: including the distributive property, and more - Arithmetic: basic operations, number theory, and calculations Get AI Math Solver Question Helper to revolutionize study, solve math's problem, and to get help for your Homework. PREMIUM PLANS AVAILABLE: Get paid subscription to discover more advance features. - No Ads Anymore - Duration wise package AI Math Solver Question Helper for 1 Week - 6.99 USD Your payment will be charged to your iTunes Account as soon as your purchase is confirmed. More Details: Privacy Policy: https://damjibhaichotaliya.blogspot.com/2024/08/damjibhai-chotaliya-privacy-policy.html Terms of Use: https://damjibhaichotaliya.blogspot.com/2024/08/damjibhai-chotaliya-terms-of-use.html

App Privacy

The developer, Damjibhai Chotaliya , indicated that the app’s privacy practices may include handling of data as described below. For more information, see the developer’s privacy policy .

Data Used to Track You

The following data may be used to track you across apps and websites owned by other companies:

• Identifiers

Data Not Linked to You

The following data may be collected but it is not linked to your identity:

• Diagnostics

Privacy practices may vary, for example, based on the features you use or your age. Learn More

Information

English, Arabic, Dutch, French, German, Hindi, Indonesian, Italian, Japanese, Korean, Malay, Portuguese, Punjabi, Russian, Simplified Chinese, Spanish, Thai, Turkish, Vietnamese

• AI Math for 1 Week $6.99
• Developer Website
• App Support
• Privacy Policy

More By This Developer

Wifi Analyzer- Speed Test & QR

Introducing Qwen2-Math

GITHUB HUGGING FACE MODELSCOPE DISCORD

🚨 This model mainly supports English. We will release bilingual (English and Chinese) math models soon.

Introduction #

Over the past year, we have dedicated significant effort to researching and enhancing the reasoning capabilities of large language models, with a particular focus on their ability to solve arithmetic and mathematical problems. Today, we are delighted to introduce a series of math-specific large language models of our Qwen2 series, Qwen2-Math and Qwen2-Math-Instruct-1.5B/7B/72B. Qwen2-Math is a series of specialized math language models built upon the Qwen2 LLMs, which significantly outperforms the mathematical capabilities of open-source models and even closed-source models (e.g., GPT-4o). We hope that Qwen2-Math can contribute to the community for solving complex mathematical problems.

We evaluate our math-specific models on a series of math benchmarks. The results below demonstrate that our largest math-specific model Qwen2-Math-72B-Instruct outperforms the state-of-the-art models, including GPT-4o, Claude-3.5-Sonnet, Gemini-1.5-Pro, and Llama-3.1-405B.

Qwen2-Math: Base Models #

The base models of Qwen2-Math are initialized with Qwen2-1.5B/7B/72B, and then pretrained on a meticulously designed Mathematics-specific Corpus. This corpus contains large-scale high-quality mathematical web texts, books, codes, exam questions, and mathematical pre-training data synthesized by Qwen2.

We evaluate our Qwen2-Math base models on three widely used English math benchmarks GSM8K, Math, and MMLU-STEM. In addition, we also evaluate three Chinese math benchmarks CMATH, GaoKao Math Cloze, and GaoKao Math QA. All evaluations are tested with few-shot chain-of-thought prompting.

Qwen2-Math-Instruct: Instruction-Tuned Models #

We first trained a math-specific reward model based on Qwen2-Math-72B. We then combined this dense reward signal with a binary signal indicating whether the model answered correctly. This combined signal is used as supervision for constructing the SFT data through Rejection Sampling and also in the reinforcement learning with Group Relative Policy Optimization (GRPO) after SFT.

We evaluate Qwen2-Math-Instruct on mathematical benchmarks in both English and Chinese. In addition to the widely-used benchmarks, such as GSM8K and Math, we also involve more exams that are much challenging to fully inspect the capabilities of Qwen2-Math-Instruct, such as OlympiadBench, CollegeMath, GaoKao, AIME2024, and AMC2023. For Chinese mathematical benchmarks, we use CMATH, Gaokao (Chinese college entrance examination 2024), and CN Middle School 24 (China High School Entrance Examination 2024).

We report greedy , Maj@8 and RM@8 performance on all benchmarks in the zero-shot setting, except for the multi-choice benchmarks (including MMLU STEM and multiple-choice problems in GaoKao and CN Middle School 24) with a 5-shot setting. Qwen2-Math-Instruct achieves the best performance among models of the same size, with RM@8 outperforming Maj@8, particularly in the 1.5B and 7B models. This demonstrates the effectiveness of our math reward model.

In more complex mathematical competition evaluations such as AIME 2024 and AMC 2023, Qwen2-Math-Instruct also performs well across various settings, including Greedy, Maj@64, RM@64, and RM@256.

Case Study #

Here we list some test cases, which include some IMO math problems. From the experimental results and case study, we find that Qwen2-Math is capable of solving simple math competition problems. Feel free to click the expandable blocks to check the cases!

All the solutions are generated by our model without modification. Please note that we do not guarantee the correctness of the claims in the process.

What is the smallest positive integer $t$ such that there exist integers $x_1,x_2,\ldots,x_t$ with

$$ x^3_1+x^3_2+,\ldots,+x^3_t=2002^{2002}, $$

Qwen2-Math-72B-Instruct Solution

To determine the smallest positive integer $ t $ such that there exist integers $ x_1, x_2, \ldots, x_t $ satisfying the equation

$$ x_1^3 + x_2^3 + \cdots + x_t^3 = 2002^{2002}, $$

we start by analyzing the properties of cubes and their sums. We need to find the smallest $ t $ for which the sum of $ t $ cubes equals $ 2002^{2002} $.

First, we note that $ 2002 $ can be factored into prime factors:

$$ 2002 = 2 \times 7 \times 11 \times 13. $$

$$ 2002^{2002} = (2 \times 7 \times 11 \times 13)^{2002} = 2^{2002} \times 7^{2002} \times 11^{2002} \times 13^{2002}. $$

Next, we consider the properties of cubes modulo 9. Cubes modulo 9 can only be congruent to $ 0, 1, $ or $ -1 $. This is because:

$$ \begin{aligned} 0^3 &\equiv 0 \pmod{9}, \\\ 1^3 &\equiv 1 \pmod{9}, \\\ 2^3 &\equiv 8 \equiv -1 \pmod{9}, \\\ 3^3 &\equiv 27 \equiv 0 \pmod{9}, \\\ 4^3 &\equiv 64 \equiv 1 \pmod{9}, \\\ 5^3 &\equiv 125 \equiv -1 \pmod{9}, \\\ 6^3 &\equiv 216 \equiv 0 \pmod{9}, \\\ 7^3 &\equiv 343 \equiv 1 \pmod{9}, \\\ 8^3 &\equiv 512 \equiv -1 \pmod{9}. \end{aligned} $$

Now, we need to determine $ 2002^{2002} \mod 9 $. First, we find $ 2002 \mod 9 $:

$$ 2002 \div 9 = 222 \text{ remainder } 4 \quad \Rightarrow \quad 2002 \equiv 4 \pmod{9}. $$

$$ 2002^{2002} \equiv 4^{2002} \pmod{9}. $$

Next, we use Euler’s theorem, which states that $ a^{\phi(n)} \equiv 1 \pmod{n} $ for $ \gcd(a, n) = 1 $. Here, $ \phi(9) = 6 $, so:

$$ 4^6 \equiv 1 \pmod{9}. $$

We can reduce the exponent $ 2002 \mod 6 $:

$$ 2002 \div 6 = 333 \text{ remainder } 4 \quad \Rightarrow \quad 2002 \equiv 4 \pmod{6}. $$

$$ 4^{2002} \equiv 4^4 \pmod{9}. $$

$$ 4^4 = 256. $$

Finally, we find $ 256 \mod 9 $:

$$ 256 \div 9 = 28 \text{ remainder } 4 \quad \Rightarrow \quad 256 \equiv 4 \pmod{9}. $$

$$ 2002^{2002} \equiv 4 \pmod{9}. $$

We need to find the smallest $ t $ such that the sum of $ t $ cubes is congruent to 4 modulo 9. The possible sums of cubes modulo 9 are:

$$ \begin{aligned} 0 &\equiv 0 \pmod{9}, \\\ 1 &\equiv 1 \pmod{9}, \\\ -1 &\equiv 8 \pmod{9}, \\\ 1 + 1 &\equiv 2 \pmod{9}, \\\ 1 + (-1) &\equiv 0 \pmod{9}, \\\ -1 + (-1) &\equiv -2 \equiv 7 \pmod{9}, \\\ 1 + 1 + 1 &\equiv 3 \pmod{9}, \\\ 1 + 1 + (-1) &\equiv 1 \pmod{9}, \\\ 1 + (-1) + (-1) &\equiv -1 \equiv 8 \pmod{9}, \\\ -1 + (-1) + (-1) &\equiv -3 \equiv 6 \pmod{9}, \\\ 1 + 1 + 1 + 1 &\equiv 4 \pmod{9}. \end{aligned} $$

We see that the sum of four cubes can be congruent to 4 modulo 9. Therefore, the smallest $ t $ is 4.

Thus, the smallest positive integer $ t $ such that there exist integers $ x_1, x_2, \ldots, x_t $ with

$$ x_1^3 + x_2^3 + \cdots + x_t^3 = 2002^{2002} $$

$$ \boxed{4}. $$

For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $ 0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?

To determine whether it is possible for $ s(k) $ and $ s(P(k)) $ to have the same parity for all positive integers $ k $, we need to analyze the behavior of the sum of the digits function $ s $ and the polynomial $ P(x) $.

First, recall that the sum of the digits of a number $ k $ modulo 2 is equivalent to $ k \mod 2 $. This is because the sum of the digits of $ k $ in base 10 is congruent to $ k \mod 9 $, and since $ 9 \equiv 1 \mod 2 $, the sum of the digits modulo 2 is the same as $ k \mod 2 $.

Given the polynomial $ P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 $, we need to determine the parity of $ P(k) $ for any positive integer $ k $. Specifically, we need to check if $ P(k) \mod 2 $ is the same as $ k \mod 2 $.

Let’s consider the polynomial modulo 2:

$$ P(x) \equiv x^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0 \mod 2. $$

Since $ a_i $ are positive integers, $ a_i \mod 2 $ is either 0 or 1. Let $ b_i = a_i \mod 2 $. Then we can rewrite the polynomial modulo 2 as:

$$ P(x) \equiv x^n + b_{n-1}x^{n-1} + \cdots + b_1x + b_0 \mod 2. $$

Now, we need to check the parity of $ P(k) $ for $ k \equiv 0 \mod 2 $ and $ k \equiv 1 \mod 2 $.

Case 1: $ k \equiv 0 \mod 2 $

$$ P(0) \equiv b_0 \mod 2. $$

For $ s(P(0)) $ to have the same parity as $ s(0) $, $ b_0 $ must be 0. Therefore, $ a_0 $ must be even.

Case 2: $ k \equiv 1 \mod 2 $

$$ P(1) \equiv 1^n + b_{n-1} \cdot 1^{n-1} + \cdots + b_1 \cdot 1 + b_0 \equiv 1 + b_{n-1} + \cdots + b_1 + b_0 \mod 2. $$

For $ s(P(1)) $ to have the same parity as $ s(1) $, $ 1 + b_{n-1} + \cdots + b_1 + b_0 $ must be odd. Since $ 1 $ is odd, the sum $ b_{n-1} + \cdots + b_1 + b_0 $ must be even.

From the above cases, we have:

• $ a_0 $ must be even.
• The sum $ b_{n-1} + \cdots + b_1 + b_0 $ must be even.

However, if $ a_0 $ is even, then $ b_0 = 0 $. This means the sum $ b_{n-1} + \cdots + b_1 $ must be even. But since $ b_i $ are either 0 or 1, the sum of $ b_{n-1} + \cdots + b_1 $ being even implies that there must be an even number of 1’s among $ b_{n-1}, \ldots, b_1 $.

Now, consider the polynomial $ P(x) = x^n + 1 $. For this polynomial:

• $ a_0 = 1 $ (odd), which contradicts our requirement that $ a_0 $ must be even.
• The sum $ b_{n-1} + \cdots + b_1 + b_0 = 1 $ (odd), which also contradicts our requirement that the sum must be even.

Therefore, it is impossible for $ s(k) $ and $ s(P(k)) $ to have the same parity for all positive integers $ k $.

Thus, the final answer is:

$$ \boxed{\text{No}} $$

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for each $x \in \mathbb{R}^+$, there is exactly one $y \in \mathbb{R}^+$ satisfying

$$ xf(y)+yf(x) \leq 2 $$

To find all functions $ f: \mathbb{R}^+ \to \mathbb{R}^+ $ such that for each $ x \in \mathbb{R}^+ $, there is exactly one $ y \in \mathbb{R}^+ $ satisfying

$$ xf(y) + yf(x) \leq 2, $$

we will proceed step by step.

Step 1: Analyze the given inequality Given $ x \in \mathbb{R}^+ $, there exists a unique $ y \in \mathbb{R}^+ $ such that

$$ xf(y) + yf(x) \leq 2. $$

We need to explore the implications of this condition.

Step 2: Consider the case $ x = y $ Substitute $ y = x $ into the inequality:

$$ xf(x) + xf(x) = 2xf(x) \leq 2. $$

This simplifies to:

$$ xf(x) \leq 1. $$

Thus, for all $ x \in \mathbb{R}^+ $,

$$ f(x) \leq \frac{1}{x}. $$

Step 3: Consider the uniqueness condition For each $ x \in \mathbb{R}^+ $, there is exactly one $ y \in \mathbb{R}^+ $ such that $ xf(y) + yf(x) \leq 2 $. We need to ensure that this $ y $ is unique.

Step 4: Test the function $ f(x) = \frac{1}{x} $ Consider the function $ f(x) = \frac{1}{x} $. Substitute $ f(x) = \frac{1}{x} $ and $ f(y) = \frac{1}{y} $ into the inequality:

$$ x \cdot \frac{1}{y} + y \cdot \frac{1}{x} = \frac{x}{y} + \frac{y}{x}. $$

$$ \frac{x}{y} + \frac{y}{x} \leq 2. $$

Using the AM-GM inequality, we know:

$$ \frac{x}{y} + \frac{y}{x} \geq 2. $$

Equality holds if and only if $ \frac{x}{y} = 1 $, which implies $ x = y $.

Thus, for $ f(x) = \frac{1}{x} $, the inequality $ xf(y) + yf(x) \leq 2 $ holds if and only if $ x = y $. This satisfies the condition that there is exactly one $ y $ for each $ x $.

Step 5: Verify if other functions can satisfy the condition Suppose there exists another function $ g: \mathbb{R}^+ \to \mathbb{R}^+ $ such that $ g(x) \neq \frac{1}{x} $ for some $ x $. Then, for some $ x $, $ g(x) < \frac{1}{x} $ or $ g(x) > \frac{1}{x} $.

• If $ g(x) < \frac{1}{x} $, then $ xg(y) + yg(x) $ might not satisfy the inequality for all $ y $.
• If $ g(x) > \frac{1}{x} $, then $ xg(y) + yg(x) $ might exceed 2 for some $ y $.

Thus, any function other than $ f(x) = \frac{1}{x} $ would not satisfy the uniqueness condition.

Conclusion The only function $ f: \mathbb{R}^+ \to \mathbb{R}^+ $ that satisfies the given condition is:

$$ \boxed{f(x) = \frac{1}{x}}. $$

Let $Z$ be the set of all integers. Find all the function $f: Z \to Z$ such that

$$ f(4x+3y)=f(3x+y)+f(x+2y) $$

For all integers $x,y$

To find all functions $ f: \mathbb{Z} \to \mathbb{Z} $ that satisfy the functional equation

$$ f(4x + 3y) = f(3x + y) + f(x + 2y) $$

for all integers $ x $ and $ y $, we will proceed step by step.

Step 1: Initial Substitution

First, let’s substitute specific values for $ x $ and $ y $ to gain insight into the function.

Substitution 1: $ x = 0 $

$$ f(4 \cdot 0 + 3y) = f(3 \cdot 0 + y) + f(0 + 2y) $$

$$ f(3y) = f(y) + f(2y) $$

Substitution 2: $ y = 0 $

$$ f(4x + 3 \cdot 0) = f(3x + 0) + f(x + 2 \cdot 0) $$

$$ f(4x) = f(3x) + f(x) $$

Step 2: Exploring Further Substitutions

We now have two equations:

• $ f(3y) = f(y) + f(2y) $
• $ f(4x) = f(3x) + f(x) $

Substitution 3: $ y = 1 $ in the first equation

$$ f(3 \cdot 1) = f(1) + f(2 \cdot 1) $$

$$ f(3) = f(1) + f(2) $$

Substitution 4: $ x = 1 $ in the second equation

$$ f(4 \cdot 1) = f(3 \cdot 1) + f(1) $$

$$ f(4) = f(3) + f(1) $$

Step 3: Using the Equations to Find a Pattern

Substitute $ f(3) $ from the first equation into the second equation:

$$ f(4) = (f(1) + f(2)) + f(1) $$

$$ f(4) = 2f(1) + f(2) $$

Step 4: Generalizing the Function

We hypothesize that $ f $ might be a linear function. Let’s assume $ f(x) = kx $ for some integer $ k $.

Verification Substitute $ f(x) = kx $ into the original functional equation:

$$ f(4x + 3y) = k(4x + 3y) $$

$$ f(3x + y) = k(3x + y) $$

$$ f(x + 2y) = k(x + 2y) $$

The original equation becomes:

$$ k(4x + 3y) = k(3x + y) + k(x + 2y) $$

$$ 4kx + 3ky = 3kx + ky + kx + 2ky $$

$$ 4kx + 3ky = 4kx + 3ky $$

This equation holds true for all $ x $ and $ y $, confirming that $ f(x) = kx $ is a solution.

The only functions $ f: \mathbb{Z} \to \mathbb{Z} $ that satisfy the given functional equation are of the form $ f(x) = kx $ for some integer $ k $.

$$ \boxed{f(x) = kx \text{ for some integer } k} $$

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$ f(f(x)+y)+xf(y)=f(xy+y)+f(x) $$

Qwen2-Math-72B-Instruct Solution:

To find all functions $ f: \mathbb{R} \rightarrow \mathbb{R} $ that satisfy the functional equation

$$ f(f(x) + y) + x f(y) = f(xy + y) + f(x), $$

First, let’s substitute $ y = 0 $ into the equation:

$$ f(f(x) + 0) + x f(0) = f(x \cdot 0 + 0) + f(x). $$

$$ f(f(x)) + x f(0) = f(0) + f(x). $$

Rearranging terms, we get:

$$ f(f(x)) = f(x) + f(0) - x f(0). $$

Step 2: Analyzing the Simplified Equation

Let $ c = f(0) $. The equation becomes:

$$ f(f(x)) = f(x) + c - x c. $$

Step 3: Substituting $ x = 0 $

Next, substitute $ x = 0 $ into the original equation:

$$ f(f(0) + y) + 0 \cdot f(y) = f(0 \cdot y + y) + f(0). $$

$$ f(c + y) = f(y) + c. $$

Step 4: Substituting $ y = -c $

Substitute $ y = -c $ into the equation $ f(c + y) = f(y) + c $:

$$ f(c - c) = f(-c) + c. $$

$$ f(0) = f(-c) + c. $$

Since $ f(0) = c $, we have:

$$ c = f(-c) + c. $$

Subtracting $ c $ from both sides, we get:

$$ f(-c) = 0. $$

Step 5: Substituting $ y = -f(x) $

Substitute $ y = -f(x) $ into the original equation:

$$ f(f(x) - f(x)) + x f(-f(x)) = f(x(-f(x)) - f(x)) + f(x). $$

$$ f(0) + x f(-f(x)) = f(-x f(x) - f(x)) + f(x). $$

$$ c + x f(-f(x)) = f(-x f(x) - f(x)) + f(x). $$

Step 6: Considering $ f(x) = 0 $

Assume $ f(x) = 0 $ for all $ x $. Substitute $ f(x) = 0 $ into the original equation:

$$ f(0 + y) + x f(y) = f(xy + y) + f(x). $$

$$ f(y) + 0 = f(y) + 0, $$

which is always true.

Step 7: Verifying $ f(x) = x $

Assume $ f(x) = x $. Substitute $ f(x) = x $ into the original equation:

$$ f(x + y) + x f(y) = f(xy + y) + f(x). $$

$$ x + y + xy = xy + y + x, $$

The only functions that satisfy the given functional equation are $ f(x) = 0 $ and $ f(x) = x $. Therefore, the solutions are:

$$ \boxed{f(x) = 0 \text{ or } f(x) = x}. $$

A long time ago, there existed Martians with $ 3$ different colours: red, green and blue. As Mars was devastated by an intergalactic war, only $ 2$ Martians of each colours survived. In order to reconstruct the Martian population, they decided to use a machine that transforms two Martians of distinct colours into four Martians of colour different to the two initial ones. For example, if a red Martian and a blue Martian use the machine, they’ll be transformed into four green Martians.

a) Is it possible that, after using that machine finitely many times, we have $ 2022 $ red Martians, $ 2022$ green Martians and $ 2022$ blue Martians?

b) Is it possible that, after using that machine finitely many times, we have $ 2021$ red Martians, $ 2022$ green Martians and $ 2023$ blue Martians?

Let’s analyze the problem step by step.

We start with 2 red, 2 green, and 2 blue Martians. The machine transforms two Martians of distinct colors into four Martians of the third color. We need to determine if it is possible to end up with 2022 red, 2022 green, and 2022 blue Martians.

First, let’s consider the parity (odd or even nature) of the number of Martians of each color. Initially, we have:

• 2 red (even)
• 2 green (even)
• 2 blue (even)

Each transformation changes the parity of the number of Martians of the two initial colors and the parity of the number of Martians of the third color. Specifically:

• If we transform 1 red and 1 green Martian, we get 4 blue Martians. This changes the parity of red and green Martians from even to odd, and the parity of blue Martians from even to odd.
• If we transform 1 red and 1 blue Martian, we get 4 green Martians. This changes the parity of red and blue Martians from even to odd, and the parity of green Martians from even to odd.
• If we transform 1 green and 1 blue Martian, we get 4 red Martians. This changes the parity of green and blue Martians from even to odd, and the parity of red Martians from even to odd.

In all cases, the parity of the number of Martians of each color changes from even to odd or from odd to even. Therefore, after any number of transformations, the parity of the number of Martians of each color will always be the same as it was initially.

Since initially all three colors have an even number of Martians, after any number of transformations, the number of Martians of each color will remain even.

Given that 2022 is an even number, it is possible to have 2022 red, 2022 green, and 2022 blue Martians after a finite number of transformations.

Thus, the answer to part (a) is:

$$ \boxed{\text{Yes}} $$

We start again with 2 red, 2 green, and 2 blue Martians. We need to determine if it is possible to end up with 2021 red, 2022 green, and 2023 blue Martians.

As in part (a), we consider the parity of the number of Martians of each color. Initially, we have:

Each transformation changes the parity of the number of Martians of the two initial colors and the parity of the number of Martians of the third color. Therefore, after any number of transformations, the parity of the number of Martians of each color will always be the same as it was initially.

Given that 2021 is an odd number, it is impossible to have 2021 red Martians after any number of transformations. Similarly, 2023 is an odd number, making it impossible to have 2023 blue Martians.

Thus, the answer to part (b) is:

Joe and Penny play a game. Initially there are $ 5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $ k$-th move, any number of stones between $ 1$ and $ k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?

To determine the winner of the game, we need to analyze the strategy and the sequence of moves. Let’s break down the problem step by step.

Understanding the Game Rules:

• Initially, there are 5000 stones.
• Joe makes the first move (odd-numbered move), Penny makes the second move (even-numbered move), and so on.
• On the $ k $-th move, a player can remove between 1 and $ k $ stones.
• The player who removes the last stone wins.

Key Insight:

• The game is symmetric in the sense that the number of stones a player can remove on their turn increases linearly with the move number.
• The total number of moves in the game is equal to the number of stones initially in the pile, which is 5000.

Total Number of Moves:

• Since there are 5000 stones, there will be 5000 moves in total.
• Joe makes the odd-numbered moves: 1st, 3rd, 5th, …, 4999th.
• Penny makes the even-numbered moves: 2nd, 4th, 6th, …, 5000th.

Determining the Winner:

• The last move (5000th move) is an even-numbered move.
• Therefore, Penny will make the last move.

Conclusion:

• Since Penny makes the last move, she will be the one to remove the last stone.
• Hence, Penny wins the game if both players play perfectly.

$$ \boxed{\text{Penny}} $$

The school offers 10 elective courses, and each student can enroll in any number of courses. The director selects $k$ students, where although each student’s combination of courses is different, any two students have at least one course in common. At this point, it is found that any student outside these $k$ students cannot be classmates with these $k$ students regardless of how they enroll (having one course in common is enough to be classmates). Find the maximum number of $k$.

To solve this problem, we need to determine the maximum number of students $ k $ such that each student has a unique combination of elective courses, any two students share at least one course, and no student outside these $ k $ students can be classmates with all of them. This means that the set of students chosen must cover all possible courses in such a way that any new student would have to share a course with at least one of the chosen students.

First, let’s consider the total number of possible combinations of courses. Since there are 10 elective courses and each student can either take or not take each course, there are $ 2^{10} = 1024 $ possible combinations of courses. However, one of these combinations is the empty set (the student takes no courses), which is not a valid combination for our problem because it would not share any courses with any other student. Therefore, there are $ 1024 - 1 = 1023 $ non-empty combinations of courses.

We need to find the maximum number of students $ k $ such that any two students share at least one course, and no student outside these $ k $ students can be classmates with all of them. This is equivalent to finding the maximum number of vertices in a graph where each vertex represents a non-empty combination of courses, and there is an edge between two vertices if and only if the corresponding combinations share at least one course. The condition that no student outside these $ k $ students can be classmates with all of them means that the set of $ k $ students forms a dominating set in the graph.

The maximum size of such a set is the complement of the minimum size of an independent set in the graph. An independent set in this graph is a set of vertices such that no two vertices are adjacent, which means no two combinations share any course. The maximum size of an independent set in this graph is the number of combinations where each course is either taken or not taken by exactly one student, which is $ 2^9 = 512 $ (since we can choose to include or exclude each of the 9 remaining courses for each of the 10 courses).

Therefore, the minimum size of an independent set is 512, and the maximum size of a dominating set (which is the complement of the independent set) is $ 1023 - 512 + 1 = 511 $. However, we need to add 1 because the empty set is not included, so the correct calculation is $ 1023 - 511 = 512 $.

Thus, the maximum number of students $ k $ is $\boxed{512}$.

Let $ABC$ be a triangle with $\angle A = 90^{\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\angle ABD = \angle DBC$ and $\angle ACE = \angle ECB$. Segments $BD$ and $CE$ meet at $I$. Determine whether or not it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, $IE$ to all have integer lengths.

To determine whether it is possible for segments $AB$, $AC$, $BI$, $ID$, $CI$, and $IE$ to all have integer lengths in triangle $ABC$ with $\angle A = 90^\circ$, we start by analyzing the given conditions and properties of the triangle.

• $\angle A = 90^\circ$
• $\angle ABD = \angle DBC$
• $\angle ACE = \angle ECB$
• Segments $BD$ and $CE$ intersect at $I$

First, note that since $\angle A = 90^\circ$, triangle $ABC$ is a right triangle with $A$ as the right angle. Points $D$ and $E$ are such that $BD$ and $CE$ are angle bisectors of $\angle ABC$ and $\angle ACB$, respectively. Therefore, $I$ is the incenter of triangle $ABC$.

Step 1: Coordinates and Lengths Assume $A$ is at the origin $(0, 0)$, $B$ is at $(a, 0)$, and $C$ is at $(0, b)$. The lengths of the sides are:

• $BC = \sqrt{a^2 + b^2}$

Step 2: Incenter Coordinates The incenter $I$ of a right triangle with legs $a$ and $b$ and hypotenuse $\sqrt{a^2 + b^2}$ is located at:

$$ I = \left( \frac{a}{2}, \frac{b}{2} \right) $$

Step 3: Lengths of Segments $BI$ and $CI$ Using the distance formula, we calculate $BI$ and $CI$:

$$ BI = \sqrt{\left( a - \frac{a}{2} \right)^2 + \left( 0 - \frac{b}{2} \right)^2} = \sqrt{\left( \frac{a}{2} \right)^2 + \left( \frac{b}{2} \right)^2} = \frac{\sqrt{a^2 + b^2}}{2} $$

$$ CI = \sqrt{\left( 0 - \frac{a}{2} \right)^2 + \left( b - \frac{b}{2} \right)^2} = \sqrt{\left( \frac{a}{2} \right)^2 + \left( \frac{b}{2} \right)^2} = \frac{\sqrt{a^2 + b^2}}{2} $$

Step 4: Lengths of Segments $ID$ and $IE$ Since $D$ and $E$ are points on $AC$ and $AB$ respectively, and $I$ is the incenter, $ID$ and $IE$ are the distances from the incenter to the sides of the triangle. For a right triangle, these distances are given by:

$$ ID = IE = r = \frac{a + b - \sqrt{a^2 + b^2}}{2} $$

Step 5: Integer Lengths For $AB$, $AC$, $BI$, $ID$, $CI$, and $IE$ to all be integers, the following must hold:

• $a$ and $b$ are integers
• $\frac{\sqrt{a^2 + b^2}}{2}$ is an integer
• $\frac{a + b - \sqrt{a^2 + b^2}}{2}$ is an integer

Step 6: Analysis For $\frac{\sqrt{a^2 + b^2}}{2}$ to be an integer, $\sqrt{a^2 + b^2}$ must be an even integer. Let $\sqrt{a^2 + b^2} = 2k$ for some integer $k$. Then:

$$ a^2 + b^2 = 4k^2 $$

For $\frac{a + b - \sqrt{a^2 + b^2}}{2}$ to be an integer, $a + b - 2k$ must be even. Since $a$ and $b$ are integers, $a + b$ must be even.

Conclusion Given the constraints, it is possible to find integer values for $a$ and $b$ such that all the conditions are satisfied. For example, if $a = 3$ and $b = 4$, then:

• $BI = CI = \frac{5}{2} = 2.5$ (not an integer)

Thus, it is not possible for all segments $AB$, $AC$, $BI$, $ID$, $CI$, and $IE$ to have integer lengths simultaneously.

Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle ${\angle{CGH}}$.

To solve for the angle $\angle CGH$ in the given triangle $\triangle ABC$ with altitudes $AD$, $BF$, and $CE$ intersecting at the orthocenter $H$, and a line through $D$ parallel to $AB$ intersecting $EF$ at $G$, we will follow these steps:

Identify Key Points and Properties:

• $D$ is the foot of the altitude from $A$ to $BC$.
• $E$ is the foot of the altitude from $B$ to $AC$.
• $F$ is the foot of the altitude from $C$ to $AB$.
• $H$ is the orthocenter of $\triangle ABC$.
• $G$ is the intersection of the line through $D$ parallel to $AB$ and the line $EF$.

Use the Property of the Orthocenter:

• The orthocenter $H$ lies on the altitudes $AD$, $BF$, and $CE$.
• The points $D$, $E$, and $F$ are collinear with $H$ on the respective altitudes.

Analyze the Parallel Line:

• Since $DG \parallel AB$, $\angle ADG = \angle DAB$ (corresponding angles).

Consider the Cyclic Quadrilateral:

• The quadrilateral $AEHF$ is cyclic because $\angle AEH = \angle AFH = 90^\circ$.
• Therefore, $\angle EHF = 180^\circ - \angle EAF$.

Use the Parallel Line Property Again:

• Since $DG \parallel AB$, $\angle DGE = \angle BAE$ (alterFnate interior angles).

Relate Angles in Cyclic Quadrilateral:

• In $\triangle AEF$, $\angle EAF = \angle EHF$ because they are opposite angles in the cyclic quadrilateral $AEHF$.

Determine $\angle CGH$:

• Since $G$ lies on $EF$ and $DG \parallel AB$, $\angle DGE = \angle BAE$.
• $\angle BAE = \angle EAF$ (as established).
• Therefore, $\angle DGE = \angle EAF$.

Final Angle Calculation:

• Since $\angle EAF = \angle EHF$ and $\angle EHF = 90^\circ$ (as $H$ is the orthocenter and $E$ and $F$ are feet of perpendiculars), we have: $$ \angle CGH = \angle EHF = 90^\circ. $$

Thus, the angle $\angle CGH$ is $\boxed{90^\circ}$.

Decontamination #

We conduct decontamination methods on both our pretraining and post-training datasets. Specifically, for pretraining data, we target on math datasets, including GSM8K, MATH, and remove samples that have significant overlaps with the test sets. We use exact match to remove the identical samples and further apply 13-gram deduplication (with a condition that the ratio of longest common sequence should be larger than 0.6) to remove more samples that might cause contamination. For post-training data, we remove more postitive contaminated samples that have overlaps with GSM8K, MATH, Aqua, SAT Math, OlympiadBench, College Math, AIME24, AMC23, etc. with the same filtering method.

This time, we’re releasing a new model series focused on mathematical capabilities, Qwen2-Math, built upon the Qwen2 foundation. Our flagship model, Qwen2-Math-72B-Instruct, outperforms proprietary models such as GPT-4o and Claude 3.5 in math-related tasks. Given the current limitation of English-only support, we plan to release bilingual models that support both English and Chinese shortly, with the development of multilingual models also in the pipeline. Moreover, we will continue to enhance our models’ ability to solve complex and challenging mathematical problems.