# Practice for the AMC8 and MATHCOUNTS Competitions

In the Brilliant course, Math Competition Fundamentals all of the math you need to learn for math competitions like MATHCOUNTS and the AMC8 is sorted into nine units that span four categories of mathematics: algebra, geometry, combinatorics, and number theory. This page compiles all of the printable practice tests (two for each unit), and also the syllabus for each unit in this course.

#### Contents

## Practice Quizzes

These practice quizzes combine and review all of the techniques used in the units in the ways that they would appear in a math competition. The answer formats to different competitions vary, so we have included both open-ended and multiple choice responses. To replicate a real math competition:

Print out the problems in these quizzes using the links provided below. This way, you won't know any of the answers until you check them all using the Brilliant website.

Math competitions provide, on average, around 2 minutes per problem. Use a stopwatch to time how long you take on each problem.

Alternatively, you can use a timer and restrict yourself to

**30 to 40 minutes**for completing the entire practice set in one sitting.Some competitions allow students to use calculators while others do not. We encourage you to use a calculator only for the most in-depth calculations on this practice quiz.

Practice Quizzes:

- Algebra: Equations (Practice 1, Practice 2)
- Algebra: Data (Practice 1, Practice 2)
- Geometry: Measures (Practice 1, Practice 2)
- Geometry: Similarity (Practice 1, Practice 2)
- Geometry: Composites (Practice 1, Practice 2)
- Combinatorics: Counting (Practice 1, Practice 2)
- Combinatorics: Probability (Practice 1, Practice 2)
- Number Theory: Efficiency (Practice 1, Practice 2)
- Number Theory: Factorization (Practice 1, Practice 2)

## Algebra: Equations

In this unit mathletes will learn to manipulate the relationship between two quantities and use ratios, rates, percentages, and equations to solve for unknowns. The unit also focuses on improving mathletes' problem-solving intuition when applying algebraic solutions. Mathletes will learn techniques to reduce the number of steps needed to solve a problem, like carefully choosing which unknown to represent with a variable or solving for a ratio of two unknowns rather than each individually.

**GOALS**

Mathletes who complete this unit will be able to:

- Solve for unknown quantities using ratios, rates, and percentages.
- Represent ratios, rates, and percentages both visually and numerically.
- Generate algebraic expressions to represent situations, and use them to solve for an unknown.
- Describe patterns algebraically.
- Choose the best way to represent an unknown with variables to create the simplest equation possible.
- Describe the relationship between two shapes using ratios, even when exact measurements are unknown.

**GAMEPLAN**

All mathletes should begin with the two diagnostic quizzes to ensure they have the basic skills needed to complete the rest of the unit. The first diagnostic covers a basic understanding of Ratios and Percentages, while the second covers Basic Equations. Mathletes should also check the prerequisites listed below to make sure they are confident in all of them before beginning the rest of the unit content.

Mathletes can then proceed with any of the other three content quizzes. Sequences and Series applies the tools of algebra to represent unknown numbers in sequences. The focus is on thoughtfully selecting variables in ways that can dramatically shorten the amount of work needed to solve an equation. Equations with Ratios dives into more complicated equations and explores algebraic tricks that can be used to solve them efficiently. Non-numeric Geometric Ratios explores techniques to apply ratios to geometric shapes without known measurements, using algebraic formulas.

**PREREQUISITES**

All mathletes completing this unit should be able to manipulate and solve for unknowns using basic ratios, rates, and percentages. These techniques are covered in the Ratios and Percentages chapter of the Mathematical Fundamentals course as well as the Rates and Ratios chapter of the Algebra Through Puzzles course. Mathletes should also be familiar with the basic strategies for solving equations, such as the elimination and substitution of variables, covered in the Balancing Scales chapter of the Algebra Through Puzzles course.

## Algebra: Data

This unit dives into the measures of central tendency of a numerical data set. An emphasis is placed on finding creative and efficient ways to calculate these measures and apply them to complicated sets of data. Mathletes will also develop problem-solving strategies to work backwards from these measures to determine specific aspects and elements of the sets they measure.

**GOALS**

Mathletes who complete this unit will be able to:

- Calculate the arithmetic mean, median, range, and mode of a variety of sets.
- Use arithmetic shortcuts to calculate the mean of certain sets.
- Predict how a change in the elements of a set would change the measures of central tendency of the set.
- Calculate a weighted mean and use the weight to determine how a change in one element affects the mean.
- Determine an unknown element of a set based on given measures of central tendency.
- Construct all the possible values for sets with certain measures of central tendency.
- Construct all the possible values for sets with certain measures of central tendency.

**GAMEPLAN**

All mathletes should begin with the Diagnostic Quiz to ensure they have the basic skills needed to complete the rest of the unit. Mathletes should also check the prerequisites listed below to make sure they are confident in all of them before beginning the rest of the unit content.

Mathletes who are unfamiliar with the measures of central tendency or simply want more training with the basics should begin with the first quiz, Data Measures. Mathletes can then study how to change these measures in Changing Data Sets.

Mathletes who are already familiar with how to calculate these measures can proceed directly to the second half of the unit, starting with Determine the Set and ending with Multiple Possibilities. This part of the unit focuses on manipulating the definition of these data measures, so mathletes at this point should be well versed with standard ways to manipulate and solve simple algebraic equations.

**PREREQUISITES**

All mathletes who are taking this unit should be able to efficiently add, subtract, multiply, and divide numbers using strategies like those found in the quizzes Arithmetic Tricks I and Arithmetic Tricks II of the Algebra Through Puzzles course. Mathletes completing the second half of this unit must know how to manipulate and solve simple algebraic equations using steps like those described in the quiz Balancing Scales, also in the Algebra Through Puzzles course.

## Geometry: Measures

This unit pushes mathletes to build on their knowledge of basic formulas and definitions to develop a set of techniques to break down problems and solve them efficiently. Techniques will include breaking 2D and 3D shapes down into their various components, extending lines to create known shapes, and manipulating equations into recognizable forms.

**GOALS**

Mathletes who complete this unit will be able to:

- Calculate the measures of unknown angles using common angle relationships.
- Compare special right triangles (both angle-based and side-based) to similar triangles in order to efficiently solve for unknown lengths.
- Develop ways to apply the Pythagorean theorem to calculate distances in several common geometric shapes.
- Break down complex diagrams and determine when to extend lines to create new shapes and relationships
- Analyze geometric diagrams and evaluate the best way to break up or extend lines in order to efficiently calculate unknown lengths.

**GAMEPLAN**

All mathletes should begin with the Diagnostic Quiz to ensure they have the basic skills needed to complete the rest of the unit.

The quiz Angle Hunting should be completed before the quiz Polygon Angle Hunting. The first quiz introduces the most fundamental axioms and techniques to finding missing angles, and the second applies these to more complicated shapes.

The quiz Special Right Triangles should be completed before the quiz Creating Right Triangles. The former teaches specific triangles that are useful in problem-solving, while the latter develops more complex strategies to apply them to problems.

**PREREQUISITES**

Mathletes must know how to calculate common geometric measurements of 2D shapes. These are covered in the quizzes on Circles, Perimeters, and Surface Area in the Mathematical Fundamentals course and the Areas and Lengths quiz in the course Outside the Box Geometry.

Mathletes must know the relationships between complementary, supplementary, vertical, and corresponding angles, as well as the sum of the interior and exterior angles of various polygons. These topics are covered in the quizzes covering the Angle Hunting Axioms and the Internal Angles in a Polygon in the course Outside the Box Geometry.

Mathletes must known and be able to apply the Pythagorean Theorem as well as the concepts of ratios and similarity to find missing sides and angles in shapes. These strategies are taught in the quizzes on The Pythagorean Theorem, Geometric Ratios, and Similarity in the Mathematical Fundamentals course.

## Geometry: Similarity

This unit focuses on building efficient problem-solving strategies to apply to complicated, multi-step geometry problems. Students will learn to analyze complicated diagrams, separate important from unimportant information, map out a piece to find the desired measurements, and recognize creative shortcuts to simplify these problems.

**GOALS**

Mathletes who complete this unit will be able to:

- Predict how changes in the dimensions of 2D and 3D shapes affect their area and volume.
- Manipulate the relationships between similar figures to determine the measures of unknown sides and angles.
- Break down complex diagrams and determine when to extend lines to create new shapes and relationships.
- Determine the equations for lines on the coordinate plane.
- Apply the distance and midpoint formulas to calculate the location of specific points on the coordinate plane.
- Manipulate the equations of lines and circles to find points of intersection.

**GAMEPLAN**

All mathletes should begin with the Diagnostic Quiz to ensure they have the basic skills needed to complete the rest of the unit. Mathletes should also check the prerequisites listed below to make sure they are confident in all of them before beginning the rest of the unit content.

The quiz on Scaling does not require any prior work other than the prerequisites listed below and can be completed before or after the other quizzes.

Exploring Similarity and Applying Similarity should be completed in the order presented since they build off of each other. Both quizzes focus on finding unknown measurements in diagrams. Each quiz uses the strategies from the previous one and incorporates new ones to tackle more complicated diagrams.

Coordinate Geometry does not require any prior work other than the prerequisites listed below and can be completed at any point.

**PREREQUISITES**

Mathletes must know how to calculate common geometric measurements of 2D and 3D shapes. These are covered in the quizzes on Circles, Perimeters, Surface Area, and Volume in the Mathematical Fundamentals course, the 3D Composites quiz in the Geometry Fundamentals course, and the Areas and Lengths quiz in the course Outside the Box Geometry.

Mathletes must know how to manipulate proportional relationships between shapes. The techniques to do so are covered in the quizzes on Geometric Ratios, Similarity, and Scaling in the Mathematical Fundamentals course as well as the quiz on Congruent and Similar Triangles in the course Outside the Box Geometry.

Mathletes must understand the basics of mapping points and shapes on the coordinate grid. They must know how to calculate the distances between points, determine the outcome of a reflection or translation, and find the equation of perpendicular lines. These skills are covered in the section on The Coordinate Plane in the Geometry Fundamentals course.

## Geometry: Composites

In this unit mathletes will develop the problem-solving sense and strategies needed to tackle composite figures. Mathletes will learn how to break down complex shapes into combinations of familiar ones so that they can apply familiar formulas. Mathletes will also learn to apply their understanding of ratios and algebra to numerically represent the relationship between two shapes even when specific measurements are not given.

**GOALS**

Mathletes who complete this unit will be able to:

- Deconstruct complex composite figures into their basic components, and use their relationships to calculate the area and perimeter of unorthodox shapes.
- Manipulate the relationships between inscribed and circumscribed figures and use them to predict possible shapes in constrained spaces.
- Calculate the probability of an outcome geometrically.
- Mentally deconstruct 3D shapes to calculate lengths, areas, volumes, etc.

**GAMEPLAN**

All mathletes should begin with the Diagnostic Quiz to ensure they have the basic skills needed to complete the rest of the unit. Mathletes should also check the prerequisites listed below to make sure they are confident in all of them before beginning the rest of the unit content.

Mathletes should then complete the quizzes in the order they are presented. Starting with Lunes and Leaves and moving on to Inscribed Figures will help mathletes develop the basic strategies needed to break down composite shapes.

Mathletes will then apply these skills directly to relate different parts of diagrams in the quiz Ratios Meet Geometry. Finally, they will apply these techniques to three dimensional diagrams in Working in 3D.

**PREREQUISITES**

Mathletes must know how to calculate common geometric measurements of 2D and 3D shapes. These are covered in the quizzes on Circles, Perimeters, Surface Area, and Volume in the Mathematical Fundamentals course, the 3D Composites quiz in the Geometry Fundamentals course, and the Areas and Lengths quiz in the course Outside the Box Geometry.

Mathletes should also be able to set up basic ratios between two quantities. These techniques are covered in the Ratios and Percentages chapter of the Mathematical Fundamentals course as well as the Rates and Ratios chapter of the Algebra Through Puzzles course.

Finally, mathletes should be comfortable assigning a variable to an unknown and setting up an equation to solve for its value. This can be practiced in the Using Variables quiz in the Mathematical Fundamentals course and in the Balancing Scales chapter of the Algebra Through Puzzles course.

## Combinatorics: Counting

This unit introduces mathletes to structured ways to solve counting problems in math competitions. The initial quizzes introduce visual tools to arrange outcomes for easier counting, and then use arithmetic shortcuts to derive even more efficient formulas. The following quizzes build mathletes' problem-solving techniques to tackle problems involving order, symmetry, and recursion.

**GOALS**

Mathletes who complete this unit will be able to:

- Construct visual representations such as Venn and Branch diagrams to organize and count the elements in a set or subset.
- Apply the rule of product to calculate the number of ways of performing multiple actions.
- Use factorials to calculate the number of possible permutations of a group of objects.
- Identify situations with symmetrical outcomes and use the symmetry to shorten the process of counting them.
- Compare situations in which the order of elements matter and when it does not.

**GAMEPLAN**

Mathletes who want more training with the basics of counting different groups should begin with the quizzes on Venn Diagrams and Branch Diagrams. Both quizzes introduce useful visual tools for organizing and counting the elements of various sets.

Mathletes who are comfortable with these strategies can move on to the quizzes on Symmetry and Over-Counting, in either order. Both of these quizzes tackle specific types of counting problems and the techniques that can be applied to solve them.

**PREREQUISITES**

All mathletes completing this unit should understand the ways to rearrange objects or count subsets of a group using strategies like those found in the Riddles of Order and Crafty Counting quizzes in the Logic course.

Competitive math problems frequently ask questions that require mathletes to count the number of multiplies of a given number. Mathletes should know how to apply the divisibility rules found in the Digits & Divisibility chapter in the Mathematical Fundamentals course.

## Combinatorics: Probability

This unit will teach mathletes a number of strategies to calculate the probability of events in a variety of situations. The unit begins with an introduction to the fundamental idea of calculating probability based on outcomes. The following quizzes dive into important principles and strategies that will help mathletes calculate probabilities without having to count every single outcome. Mathletes will learn to use complements, symmetry, and visual representations to simplify complicated situations.

**GOALS**

Mathletes who complete this unit will be able to:

- Calculate the probability of an event based on the total and desired number of outcomes.
- Determine whether to apply the rule of sum or rule of product in a given situation.
- Break down possible outcomes and then apply the rule of complement and the principle of inclusion and exclusion to quickly calculate complex events.
- Develop an algorithm to determine the number of ways to select a given number of objects from a larger group.
- Visually represent the distribution of objects and calculate the number of possible distributions.
- Simplify probability calculations using the symmetry of outcomes.
- Represent possible outcomes to determine a probability given knowledge of another outcome.

**GAMEPLAN**

Mathletes should then complete the quiz on Probability by Outcomes to familiarize themselves with the types of situations often presented in competitive math problems.

Mathletes can then complete the quizzes on PIE and Complements, Choosing, and Symmetry and Conditional probability in whatever order they choose, as they all teach different strategies to tackle more complicated problems.

**PREREQUISITES**

All mathletes completing this unit should understand the ways to rearrange objects or count subsets of a group using strategies like those found in the Riddles of Order and Crafty Counting quizzes in the Logic course.

Mathletes should know the ways to operate with, manipulate, and compare ratios, fractions, and percentages. Many of these skills can be learned in the Ratios & Percentages chapter in the Mathematical Fundamentals course.

## Number Theory: Efficiency

This unit focuses on a variety of time-saving techniques that are crucial to mathletesâ€™ success in competitions. Mathletes will learn how to shorten calculations, simplify exponents, and approximate and evaluate square roots in order to save time for the harder problems in math competitions. Mathletes will also learn how to use basic number facts and number sense to predict the outcomes of several possible operations and only select relevant cases.

**GOALS**

Mathletes who complete this unit will be able to:

- Shorten calculations using common algorithms.
- Predict the general outcome of an operation before calculating it.
- Simplify exponents using the product and power rules.
- Manipulate expressions with exponents and roots to avoid operations.
- Apply the difference of squares algorithm to evaluate sums and products.
- Use basic number facts to break down complex number puzzles and evaluate all possible cases.

**GAMEPLAN**

Mathletes can then proceed to the quiz on Calculations or Exponents as they each teach different, unrelated strategies. Once mathletes have completed the Exponents quiz they can proceed to the Roots quiz, which builds on the strategies presented in Exponents.

The quiz Whatâ€™s the Number does not build directly on any of the skills in the other quizzes, but does require a strong overall number-sense that can be built up throughout the previous quizzes.

**PREREQUISITES**

All mathletes completing this unit should feel comfortable with basic arithmetic skills such as addition, subtraction, multiplication, division, exponents, and order of operations. They can practice these in the Basic Arithmetic practice section.

Many of the arithmetic shortcuts covered in this unit are also covered in the Simplifying Shortcuts chapter of the Algebra Through Puzzles course. While this chapter is not completely required to access the skills taught in this Competition unit, it can help mathletes develop fluency with the skills covered.

## Number Theory: Factorization

In this unit mathletes will explore the many uses of the prime factorization of an integer. Mathletes will use the prime factorization to break down numbers to find divisors and multiples and to construct numbers given certain specifications. Mathletes will then use divisibility rules and number facts to determine all the possible factors in a given multiplication problem and solve for missing digits.

**GOALS**

Mathletes who complete this unit will be able to:

- Use the prime factorization to calculate the number of divisors of an integer.
- Construct a number with a specific number of factors based on its prime factorization.
- Efficiently calculate the greatest common divisor and least common multiple of multiple numbers.
- Evaluate and simplify factorials.
- Identify key factors in a factorial expression.
- Use divisibility rules to deconstruct all possible factors and products in multiplication problems.

**GAMEPLAN**

All mathletes should begin with the Diagnostic Quiz to ensure they have the basic skills needed to complete the rest of the unit.

Mathletes should then continue with the Number of Divisors quiz which introduces the way to use the prime factorization to determine possible divisors. A similar strategy is then used in the GCD and LCM quiz as well as the Factorials quiz.

Mathletes can complete the Cryptograms quiz at any point in the sequence, but should be fully comfortable with the divisibility rules prerequisites outlined at the end of this syllabus.

**PREREQUISITES**

All mathletes completing this unit should be able to find the prime factorization of a number and compute the common divisors and multiples of a pair of integers. Mathletes can practice these skills in the quizzes on Factor Trees, Prime Factorization, the LCM, and the GCD in the Number Theory course.

Mathletes should know the common rules and shortcuts for determining whether or not a number is divisible by a given integer. Mathletes can learn and practice these rules in the DIgits & Divisibility unit in the Mathematical Fundamentals course.

**Cite as:**Practice for the AMC8 and MATHCOUNTS Competitions.

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