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Math Practice Problems with Hints and Answers
Work through original AMC-style and exam-specific practice questions organized by topic. Every page gives you the concept to use, a hint to get unstuck, and the answer so you can check your work.
400 individual problem walkthroughs are currently published.
Absolute Value (10)
- Find the distance between math expression and math expression on a number line.
- Solve math expression; find product of solutions.
- math expression passes through math expression; its positive vertex is math expression. Find math ex
- Find sum of integers satisfying math expression.
- **(Proof track.)** Give a complete justification. Find the minimum of math expression.
- **(Proof track.)** Give a complete justification. Find the maximum of math expression over real math
- **(Proof track.)** Give a complete justification. Solve math expression; enter the larger solution.
- **(Proof track.)** Give a complete justification. Find the sum of solutions of math expression.
Coordinate Geometry (20)
- What is the math expression-coordinate of the point math expression?
- Find the distance between math expression and math expression.
- A line has slope math expression and passes through math expression. Write its equation.
- Find the midpoint of math expression and math expression as an ordered pair.
- The point math expression is how far from the math expression-axis?
- Find the area of the rectangle with opposite vertices math expression and math expression.
- Find the midpoint of a segment whose endpoints are math expression and math expression.
- Reflect math expression across the math expression-axis. Give the new math expression-coordinate.
Plane Geometry (10)
- A circle has radius math expression. Find its area in terms of math expression.
- Two vertical angles measure math expression and math expression. Find math expression.
- A right angle is divided into angles of math expression and math expression. Find math expression.
- A square's side doubles from math expression to math expression. What is its new area?
- A triangle has base math expression and area math expression. Find its height.
- What is the sum of interior angles of a quadrilateral?
- A circle's circumference is math expression. Find its area.
- The exterior angle of a regular polygon is math expression. Find its number of sides.
Pre-Algebra & Number (10)
- What is the least common multiple of math expression and math expression?
- Convert math expression to a fraction in simplest form.
- A math expression dollar bill is split equally among math expression people. What does each pay?
- Order math expression, math expression, and math expression from least to greatest.
- A store has math expression boxes with math expression pencils each. After selling math expression p
- What is the greatest common factor of math expression and math expression?
- A sequence starts math expression. What is its math expressionth term?
- If math expression is odd, which is always even: math expression, math expression, or math expressio
Trigonometry Basics (10)
- A right triangle has adjacent side math expression and opposite side math expression. Find math expr
- If math expression in a right triangle, find the adjacent-to-hypotenuse ratio.
- A ladder math expression ft long reaches math expression ft up a wall. What is the sine of its angle
- If math expression and math expression is acute, find math expression.
- In triangle math expression, math expression and math expression. Find math expression.
- A ramp rises math expression ft over a horizontal run of math expression ft. Find math expression.
- A sinusoid has maximum math expression and minimum math expression. Find its amplitude.
- A point on the unit circle has coordinates math expression. What angle in degrees corresponds to it?
Rates & Word Problems (10)
- Five identical movie tickets cost math expression dollars. What is one ticket's cost?
- A worker earns math expression dollars per hour for math expression hours. How much does the worker
- A math expression dollar jacket is on sale for math expression off. What is the sale price?
- A train leaves at 2:15 PM and arrives at 5:45 PM. How long is the trip in hours?
- A recipe needs math expression eggs for math expression cakes. How many eggs are needed for math exp
- A tank contains math expression gallons and leaks math expression gallons each hour. How many gallon…
- A book has math expression pages. Maya reads math expression pages per day. How many pages remain af
- A rectangular garden is math expression m by math expression m. How many meters of fence are needed?
Derivatives (5)
- Critical points of math expression: enter the positive one.
- FTC: math expression at math expression?
- Linearization of math expression at math expression: approximate math expression as math expression
- Where math expression increases: left endpoint of increasing interval containing math expression?
- If math expression and math expression, find math expression at that point.
Differential Equations (2)
Integrals (3)
Limits (1)
Series & Sequences (3)
Area Techniques (10)
- Find the area of a triangle with base math expression and height math expression.
- Find the area of a parallelogram with base math expression and height math expression.
- Find the area of a circle with radius math expression in terms of math expression.
- Find the area of a right triangle with legs math expression and math expression.
- Find the area of a trapezoid with bases math expression and height math expression.
- Find the area of an equilateral triangle of side math expression.
- Find the area of a semicircle of radius math expression in terms of math expression.
- Find the area of a rhombus with diagonals math expression and math expression.
Arithmetic Sequences (10)
- Find the common difference of math expression.
- Find the average of the first and last terms of math expression.
- Find the sum of the first math expression terms of math expression.
- Find the sum of all multiples of math expression from math expression through math expression.
- An arithmetic sequence has math expression terms, first term math expression, last term math express
- How many terms of math expression are positive?
- If the sum of the first math expression positive integers is math expression, find math expression.
- math expression and math expression for an arithmetic sequence. Find math expression.
Base Systems (8)
- Find the least base math expression such that math expression is divisible by math expression.
- **(Proof track.)** Give a complete justification. Find base math expression if math expression.
- **(Proof track.)** Give a complete justification. Write math expression in base math expression.
- **(Proof track.)** Give a complete justification. For what base math expression is math expression a…
- **(Proof track.)** Give a complete justification. Compute math expression in base math expression.
- **(Proof track.)** Give a complete justification. Find the sum of digits of math expression in base
- **(Proof track.)** Give a complete justification. Find largest base in which math expression.
- **(Proof track.)** Give a complete justification. Evaluate math expression in base ten.
Circle Geometry (10)
- Find the circumference of a circle with radius math expression in terms of math expression.
- A central angle measures math expression. What fraction of the circle is its sector?
- An inscribed angle intercepts a math expression arc. Find the angle.
- Find the arc measure intercepted by a central angle of math expression.
- A radius is perpendicular to a chord. What does it do to the chord? Type math expression for bisects
- Find the circumference of a circle with diameter math expression in terms of math expression.
- A semicircle has arc measure how many degrees?
- A diameter subtends what angle at a point on the circle?
Combinations (10)
- How many math expression-element subsets does a math expression-element set have?
- How many ways can a committee of math expression be selected from math expression people?
- How many ways can math expression toppings be chosen from math expression?
- How many pairs can be formed from math expression people?
- How many ways can math expression cards be chosen from math expression cards?
- How many subsets of size math expression does any set have?
- How many triangles are determined by math expression points with no three collinear?
- How many ways can math expression flavors be chosen from math expression?
Divisibility (10)
- Find the missing digit math expression so math expression is divisible by math expression.
- What is the smallest positive multiple of math expression that is also divisible by math expression?
- Find the remainder when math expression is divided by math expression.
- Is math expression divisible by math expression? Type math expression for yes and math expression fo
- What digit makes math expression divisible by math expression and as small as possible?
- How many multiples of math expression are from math expression through math expression?
- Find the least number to add to math expression to make a multiple of math expression.
- Find the greatest common divisor of math expression and math expression.
Floor and Ceiling (8)
- How many integers math expression satisfy math expression and math expression?
- Find math expression.
- How many integers are strictly between math expression and math expression?
- How many nonnegative lattice points satisfy math expression?
- **(Proof track.)** Give a complete justification. How many integers satisfy math expression?
- **(Proof track.)** Give a complete justification. How many positive integers math expression have ma…
- **(Proof track.)** Give a complete justification. Find the greatest integer less than math expressio…
- **(Proof track.)** Give a complete justification. How many pairs math expression of positive integer…
Conic Sections (10)
- Ellipse math expression: math expression?
- Circle through origin radius math expression center origin: equation constant?
- Identify math expression: type — enter math expression hyperbola in rotated axes sense / rectangular
- Ellipse math expression: eccentricity math expression with math expression: math expression?
- Hyperbola math expression: asymptote slope positive?
- Normal to math expression at math expression: slope of normal?
- How many lines are tangent to both concentric circles math expression and math expression?
- Find the eccentricity of the hyperbola math expression.
Inequalities (5)
- Solve math expression: number of integer solutions?
- Find the minimum value of math expression for math expression.
- Rearrangement: sorted same order max sum products — for math expression and math expression max dot?
- Schur: for math expression, math expression vs math expression: enter left−right at equality math ex
- uvw / symmetric: for math expression, math expression, math expression: are they all math expression
Probability (13)
- A binomial random variable has math expression trials and success probability math expression. What
- A fair coin is flipped once. What is math expression?
- Assuming 365 equally likely birthdays, is the probability that at least two of 23 people share a bir…
- A card numbered math expression through math expression is chosen. What is math expression?
- Repeated: math expression?
- A bag has math expression red and math expression blue marbles. What is math expression?
- A fair die is rolled. What is math expression?
- Two fair coins are flipped. What is math expression?
Sequences & Series (2)
Solid Geometry (9)
- Cylinder math expression, math expression: volume in terms of math expression?
- Regular tetrahedron edge math expression: height math expression: enter math expression coefficient …
- Frustum — skip. Sphere diam math expression: volume math expression?
- Skew lines in cube opposite edges: distance if edge math expression?
- Sphere ∩ plane through center: circle radius if sphere math expression?
- Optimize cylinder in sphere math expression: max volume cylinder height? For math expression: height
- Truncated cone volumes — math expression: volume math expression?
- Vector: volume parallelepiped edges math expression?
Geometric Sequences (10)
- A bacteria count doubles each hour from math expression. Find the count after math expression hours.
- A geometric sequence has math expression and math expression. Find math expression.
- Find the infinite sum math expression.
- A ball bounces to half its previous height, beginning at math expression. Find its height after math
- Find the product of the first math expression terms of math expression.
- A sequence has math expression and math expression. Find math expression if the ratio is positive.
- A ball rebounds to math expression of its height. Dropped from math expression, find fourth-bounce h
- Three consecutive geometric terms have product math expression. Find the middle term.
Inclusion-Exclusion (10)
- How many integers from math expression to math expression are divisible by math expression **or** ma
- Sets math expression satisfy math expression, math expression, math expression. Find math expression
- How many integers from math expression to math expression are divisible by math expression **or** ma
- How many surjective functions are there from a math expression-element set to a math expression-elem
- How many length-math expression binary strings contain **no** two consecutive math expressions?
- How many integers from math expression to math expression are divisible by math expression **or** ma
- How many functions from a math expression-element set to a math expression-element set are **surject
- How many integers from math expression to math expression are divisible by **neither** math expressi
AM-GM Inequality (10)
- Solve math expression for the greatest integer math expression.
- If math expression, is math expression allowed? Type math expression for yes.
- For positive math expression with math expression, find the maximum possible math expression.
- For positive math expression, find the minimum of math expression.
- Solve math expression. Find the greatest integer solution.
- For positive math expression with math expression, find the minimum of math expression.
- For positive math expression with math expression, find the maximum of math expression.
- Solve math expression. Find the greatest integer math expression.
Calculus for JEE (2)
Complex Numbers (1)
Matrices & Determinants (6)
Probability & Counting (3)
Vectors (5)
- Angle between math expression and math expression in degrees?
- Normal to plane math expression: first component of a normal?
- Angle between planes math expression and math expression degrees?
- Shortest distance between skew lines? For math expression-axis and line math expression: distance?
- Line through math expression direction math expression meets plane math expression at math expressio
Modular Arithmetic (10)
- How many integers math expression satisfy math expression?
- What is the remainder when math expression is divided by math expression?
- Find the smallest positive integer math expression with math expression and math expression.
- How many integers math expression are coprime to math expression?
- Number of solutions to math expression.
- Least composite math expression with math expression?
- How many math expression are coprime to math expression?
- **(Proof track.)** Prove the Chinese Remainder Theorem for pairwise coprime moduli, then find the un…
Permutations (10)
- How many ways can the letters of math expression be arranged?
- How many math expression-digit numbers can be made from math expression without repetition?
- How many ways can you choose a president then vice president from math expression people?
- How many ways can math expression people sit in math expression labeled chairs?
- How many arrangements of math expression begin with math expression?
- How many ways can math expression different prizes be awarded to math expression people, with no rep
- How many math expression-letter strings without repetition use math expression?
- How many ways can math expression cards be ordered in a row?
Pigeonhole Principle (10)
- What is the minimum number of people needed to **guarantee** that two share a birth month?
- From a standard math expression-card deck, what is the minimum number of cards you must draw to **gu…
- Choose integers from math expression. What is the minimum size of a subset that **guarantees** two n…
- What is the minimum number of people needed to **guarantee** that two were born on the same day of t…
- Balls are placed into math expression identical boxes (each box can hold many). What is the minimum …
- Pick integers from math expression. What is the minimum size guaranteeing two **consecutive** intege…
- What is the minimum number of people needed to **guarantee** that two share the same zodiac sign (th…
- Pick integers from math expression. What is the minimum size guaranteeing two with the **same remain…
Power of a Point (10)
- Intersecting chords: segments math expression and math expression; on the other chord math expressio
- From external point, tangent length math expression and secant external segment math expression. Fin
- Chords intersect: math expression. Find math expression.
- From external point, tangent length math expression and whole secant length math expression. Find th…
- Tangent length math expression, whole secant math expression. External secant segment?
- Two circles intersect at math expression. Line through math expression on math expression meets circ…
- Point math expression w.r.t. circle math expression. Find power of the point.
- External secant: external part math expression, whole secant math expression. Tangent length?
Prime Factorization (10)
- How many distinct prime factors does math expression have?
- What is the product of the distinct prime factors of math expression?
- How many perfect-square divisors does math expression have?
- What is the least math expression such that math expression is a perfect square?
- How many positive factors of math expression are multiples of math expression?
- Find the least positive integer with exactly math expression positive divisors.
- How many positive divisors does math expression have?
- Find the least math expression such that math expression is divisible by math expression.
Quadratics (10)
- Find the vertex math expression-coordinate of math expression.
- If math expression, find math expression for the larger root.
- Find math expression if math expression has double root math expression.
- For what math expression does math expression have one real root?
- A rectangle has perimeter math expression and area math expression. Find its longer side.
- Find the sum of positive roots of math expression.
- For how many integers math expression does math expression have integer roots?
- math expression passes through math expression. Find math expression.
Data & Statistics (10)
- A bag has math expression green and math expression yellow tokens. What is the probability of drawin
- The mean of five values is math expression. What is their total?
- In a data set, math expression of math expression observations are red. How many are red?
- A spinner has math expression equal sections numbered 1 through 6. What is the probability of an eve
- What is the interquartile range if math expression and math expression?
- A fair coin is flipped twice. What is the probability of exactly one head?
- A sample of math expression voters has math expression in favor. Estimate the number in favor among …
- The values math expression have a new value math expression added. What is the new mean?
SAT Geometry (10)
- A triangle has angles math expression and math expression. Find the third angle.
- Two supplementary angles are math expression and math expression. Find math expression.
- A right triangle has legs math expression and math expression. Find its hypotenuse.
- Find the measure of each interior angle of a regular hexagon.
- A trapezoid has bases math expression and math expression and height math expression. Find its area.
- Similar triangles have side ratio math expression. If the smaller area is math expression, find the
- A rectangular prism measures math expression by math expression by math expression. Find its surface
- In a math expression-math expression-math expression triangle, the short leg is math expression. Fin
Linear Equations (10)
- A gym charges a math expression dollar signup fee plus math expression dollars per visit. Which expr…
- The line through math expression and math expression has what slope?
- For what value of math expression are math expression and math expression equivalent?
- A taxi traveled math expression miles in math expression hours at a constant rate. What was its rate
- Rewrite math expression in the form math expression. What is math expression?
- The table has math expression and math expression. What is the rate of change?
- A line is parallel to math expression and passes through math expression. What is its math expressio
- At what point do math expression and math expression intersect?
Quadratics & Parabolas (10)
- The parabola math expression opens in which direction?
- What is the axis of symmetry of math expression?
- A square has area math expression. What is its side length?
- The roots of math expression are math expression and math expression. Find math expression.
- A ball's height is math expression. At what time does it reach maximum height?
- If math expression has exactly one real root, find math expression.
- The graph of math expression is shifted right 4 and down 7. Write its equation.
- A rectangle has length math expression and width math expression. Its area is math expression. Find
Ratios & Percent (10)
- A math expression dollar shirt is discounted math expression. What is the sale price?
- A map scale is math expression inch to math expression miles. What distance does math expression inc
- A class has math expression girls and math expression boys. What is the ratio of boys to all student
- A solution has math expression cups juice and math expression cups water. What fraction of the mixtu
- If math expression notebooks cost math expression dollars, what is the unit price?
- A number changes from math expression to math expression. What is the percent increase?
- On a scale drawing, math expression cm represents math expression m. How many cm represent math expr
- A recipe for math expression muffins uses math expression cups of milk. How many cups are needed for
Similar Triangles (9)
- A math expression-foot pole casts a math expression-foot shadow. A tree casts a math expression-foot
- In similar triangles, math expression, math expression, and math expression correspond. Find math ex
- A triangle similar to a math expression-math expression-math expression triangle has hypotenuse math
- Two similar triangles have perimeters math expression and math expression. A side of the first is ma…
- A math expression by math expression rectangle is enlarged so its shorter side is math expression. F
- Triangles math expression and math expression are similar. If math expression and math expression, f
- A map scale is math expression. A road measures math expression cm on the map. How many meters is th
- A math expression-math expression-math expression triangle is reduced so its hypotenuse is math expr
Stars and Bars (9)
- How many **positive** integer solutions does math expression have?
- How many ways can math expression identical apples be split among math expression children (each may
- How many **nonnegative** solutions to math expression?
- How many **positive** solutions to math expression?
- Distribute math expression identical balls into math expression distinct bins (empty bins allowed).
- How many triples of nonnegative integers sum to math expression?
- How many ordered pairs of **positive** integers have sum math expression?
- How many **positive** solutions does math expression have?
Systems of Equations (9)
- Solve math expression, math expression for the larger possible math expression.
- Two numbers sum to math expression and differ by math expression. Find the larger.
- A math expression-dollar and a math expression-dollar item total math expression for math expression
- Chickens and rabbits total math expression heads and math expression legs. Find rabbits.
- Two positive numbers have sum math expression and difference math expression. Find product.
- Nineteen math expression- or math expression-dollar tickets cost math expression. How many are math
- **(Proof track.)** Give a complete justification. Solve math expression; find math expression.…
- **(Proof track.)** Give a complete justification. A two-digit number plus its reversal is math expre…
Triangle Inequality (9)
- Can sides math expression form a triangle? Type math expression for yes.
- Find the greatest integer third side for sides math expression and math expression.
- Find the least integer third side for sides math expression and math expression.
- A triangle has sides math expression. If math expression is an integer, find its greatest value.
- A triangle has sides math expression. Find the least integer math expression.
- How many integer values can the third side have if the other sides are math expression and math expr
- Find the perimeter of an equilateral triangle with side math expression.
- A triangle has sides math expression. Is it isosceles? Type math expression for yes.
Vieta's Formulas (9)
- Roots of math expression are math expression. Find math expression.
- For math expression, find the sum of the roots.
- For math expression, find the product of the roots.
- For math expression, find math expression where math expression are roots.
- For math expression, find the sum of squares of the roots.
- For math expression, find math expression.
- For math expression, sum of squares of all real roots.
- Sum of reciprocals of roots of math expression.
Rates and Work (9)
- At math expression dollars each, what is the cost of math expression notebooks?
- A faucet fills math expression gallons in math expression minutes. Find its rate in gallons per minu
- A runner goes math expression miles at math expression mph. Find the time in hours.
- A math expression item is discounted math expression. Find the sale price.
- A recipe for math expression uses math expression cups of flour. How many cups for math expression?
- A train travels at math expression mph for math expression hours. Find the distance.
- A worker earns math expression per hour for math expression hours. Find the pay.
- A tank has math expression liters and drains math expression liters per minute. Find the time to emp
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