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Problem Bank

104 problems, fully annotated.

Every problem carries three escalating hints, a full solution, and the mistakes students actually make. Solving updates your mastery model.

104 problems · 0 solved

48 \cdot 25 + 52 \cdot 25

.Arithmetic & Number SenseAMC 8 style~1m 1If

3x + 7 = 25

x

?Linear EquationsAMC 8 style~1m 2Compute

1 + 2 + 3 + \cdots + 40

.Arithmetic & Number SenseMathCounts style~2m 2Compute

17^2 - 13^2

.Arithmetic & Number SenseMathCounts style~1m 2What is

\tfrac{3}{4}

\tfrac{2}{5}

200

?Fractions & RatiosAMC 8 style~1m 2What is

\lfloor \sqrt{2026} \rfloor

?EstimationMathCounts style~2m 2Two angles of a triangle measure

50°

60°

. What is the third angle in degrees?AnglesAMC 8 style~1m 2A wardrobe has

4

3

pairs of pants. How many shirt-pants outfits are possible?Counting FundamentalsAMC 8 style~1m 3Compute

999 \cdot 1001

.Arithmetic & Number SenseAMC 8 style~2m 3Two numbers are in the ratio

3:5

and their sum is

96

. What is the larger number?Fractions & RatiosAMC 8 style~2m 3An

\

jacket is marked up

, then the new price is discounted

20\%$. What is the final price in dollars?Percents & RatesAMC 8 style~2m 3What is the distance between

(1, 2)

(7, 10)

?Coordinate PlaneAMC 8 style~2m 3Lines

\ell

m

are parallel. A transversal makes a

40°

\ell

. What is the measure of the co-interior (same-side interior…AnglesAMC 8 style~2m 3A right triangle has legs

9

12

. What is the length of the hypotenuse?TrianglesAMC 8 style~2m 3How many three-digit numbers have all digits odd?Counting FundamentalsAMC 8 style~2m 3How many positive divisors does

36

have?Divisibility & PrimesAMC 8 style~2m 4If

a:b = 2:3

b:c = 4:5

c = 45

a

?Fractions & RatiosAMC 8 style~3m 4Two successive

10\%

discounts are equivalent to a single discount of what percent?Percents & RatesAMC 8 style~2m 4Which of the following is the largest?EstimationAMC 10 style~3m 4Ann is twice as old as Ben. In

6

years, the sum of their ages will be

42

. How old is Ann now?Linear EquationsAMC 8 style~3m 4How many integers

n

3n - 5 < 16 < 5n - 4

?InequalitiesAMC 10 style~3m 4A line passes through

(2, 3)

(6, 11)

. If its equation is

y = mx + b

m + b

?Coordinate PlaneAMC 10 style~3m 4If

x + \tfrac{1}{x} = 5

x^2 + \tfrac{1}{x^2}

?Variables & ExpressionsAMC 10 style~3m 4If

f(x) = 2x + 3

f(f(2))

?FunctionsAMC 10 style~2m 4The roots of

x^2 - 7x + 12 = 0

r

s

r^2 + s^2

?Quadratics & VietaAMC 10 style~3m 4An arithmetic sequence has first term

5

and common difference

3

20

th term?Sequences & SeriesAMC 8 style~2m 4Each interior angle of a regular polygon measures

156°

. How many sides does it have?AnglesAMC 8 style~3m 4In a triangle, two interior angles measure

35°

65°

. What is the measure of the exterior angle at the third vertex, in degrees?AnglesAMC 8 style~2m 4The sides of a triangle are

4

9

. How many integer values can the third side take?TrianglesAMC 8 style~3m 4A

6

-foot person casts a

4

-foot shadow. At the same moment, a tree casts an

18

-foot shadow. How tall is the tree in feet?SimilarityAMC 8 style~2m 4A cube has surface area

96

. What is its volume?3D GeometryAMC 8 style~2m 4A code consists of two distinct letters (A–Z) followed by two distinct digits. How many codes are possible?Counting FundamentalsAMC 8 style~3m 4How many distinct arrangements does the word LEVEL have?Permutations & CombinationsAMC 8 style~3m 4A club of

10

students chooses a committee of

3

. How many committees are possible?Permutations & CombinationsAMC 8 style~2m 4Two fair dice are rolled. What is the probability the sum is

7

?ProbabilityAMC 8 style~2m 4What is

\gcd(84, 126)

?Divisibility & PrimesAMC 8 style~2m 4What is the remainder when

2^{10}

7

?Modular ArithmeticAMC 8 style~2m 5A chemist mixes

30

40\%

acid solution with

20

10\%

acid solution. What is the acid concentration of the mixture?Fractions & RatiosAMC 10 style~3m 5Pipe A fills a tank in

6

hours; pipe B fills it in

3

hours. Working together, how many hours do they take?Linear EquationsAMC 10 style~3m 5What is the minimum value of

f(x) = x^2 - 4x + 7

?FunctionsAMC 10 style~3m 5What is the remainder when

x^3 - 2x^2 + 3x - 4

x - 2

?PolynomialsAMC 10 style~2m 5What is the sum of the coefficients of

(2x - 1)^4

?PolynomialsAMC 10 style~2m 5If

x^2 + y^2 = 25

x + y = 7

xy

?PolynomialsAMC 10 style~3m 5If

r

s

are the roots of

x^2 - 6x + 4 = 0

\tfrac{1}{r} + \tfrac{1}{s}

?Quadratics & VietaAMC 10 style~3m 5For what positive value of

k

x^2 + kx + 9 = 0

have exactly one solution?Quadratics & VietaAMC 10 style~3m 5If

x + y = 10

x^2 - y^2 = 40

x

?SystemsAMC 10 style~3m 5An infinite geometric series has first term

6

\tfrac{1}{3}

. What is its sum?Sequences & SeriesAMC 10 style~2m 5What is the area of a triangle with sides

13

14

15

?TrianglesAMC 10 style~4m 5Two similar triangles have corresponding sides in ratio

2:5

. If the smaller has area

8

, what is the area of the larger?SimilarityAMC 10 style~3m 5A central angle of a circle measures

100°

. What is the inscribed angle subtending the same arc, in degrees?CirclesAMC 10 style~2m 5A chord of a circle of radius

10

6

from the center. How long is the chord?CirclesAMC 10 style~3m 5What is the area of the triangle with vertices

(0,0)

(8,0)

(3,7)

?Coordinate GeometryAMC 10 style~3m 5A cone has radius

6

4

. Its volume is

k\pi

k

?3D GeometryAMC 10 style~2m 5How many three-digit numbers contain at least one digit

7

?Counting FundamentalsAMC 10 style~4m 5How many integers from

1

100

are divisible by

3

5

?Casework & ComplementaryAMC 8 style~3m 5A bag has

3

2

blue marbles. Two are drawn without replacement. The probability both are red is

\tfrac{m}{n}

in lowest terms. W…ProbabilityAMC 10 style~3m 5A fair six-sided die is rolled once. If the expected value of the roll is

E

2E

?Expected ValueAMC 10 style~2m 5A drawer contains socks in

5

colors. How many socks must you draw (blindly) to guarantee a matching pair?Pigeonhole PrincipleAMC 8 style~2m 5What is the smallest positive integer with exactly

6

divisors?Divisibility & PrimesAMC 10 style~3m 5For how many primes

p

p

p+2

p+4

all prime?Divisibility & PrimesAMC 10 style~3m 5What is the units digit of

7^{2026}

?Modular ArithmeticAMC 8 style~3m 5What is the remainder when

3^{100}

8

?Modular ArithmeticAMC 10 style~2m 6Solve for

x

\log_2 x + \log_2 (x - 2) = 3

.FunctionsAMC 12 style~4m 6The cubic

x^3 - 6x^2 + 11x - 6 = 0

p, q, r

pq + qr + rp

?Quadratics & VietaAMC 12 style~3m 6If

x + y = 5

xy = 6

x^3 + y^3

?SystemsAMC 12 style~4m 6A sequence satisfies

a_1 = 1

a_n = a_{n-1} + 2n - 1

n \ge 2

a_{15}

?Sequences & SeriesAMC 10 style~4m 6An equilateral triangle has side length

12

a\sqrt{b}

b

squarefree. What is

a + b

?TrianglesAIME style~3m 6In a right triangle with legs

6

8

, the altitude to the hypotenuse has length

h

10h

?SimilarityAMC 10 style~4m 6Chords

AB

CD

of a circle meet at

P

inside the circle. If

AP = 3

PB = 8

CP = 4

PD

?CirclesAMC 10 style~3m 6Point

Q

is the reflection of

P = (3,4)

y = x

. What is the square of the distance

PQ

?Coordinate GeometryAMC 10 style~3m 6Six people sit around a round table. Two of them insist on sitting together. How many seatings are there, if rotations are considered identi…Permutations & CombinationsAMC 10 style~4m 6How many lattice paths go from

(0,0)

(4,4)

using only unit steps right and up?Permutations & CombinationsAMC 10 style~3m 6How many three-digit numbers have digits summing to

5

?Casework & ComplementaryAMC 10 style~4m 6A fair coin is flipped

4

times. The probability of at least two heads is

\tfrac{m}{n}

in lowest terms. What is

m + n

?Casework & ComplementaryAIME style~4m 6A family has two children. Given that at least one is a boy, what is the probability both are boys? (Assume boys and girls are equally likel…ProbabilityAMC 10 style~3m 6What is the smallest number of people that guarantees at least

3

of them share a birth month?Pigeonhole PrincipleAMC 10 style~3m 6Two positive integers have

\gcd = 6

\text{lcm} = 180

. If one of them is

36

, what is the other?Divisibility & PrimesAMC 10 style~3m 6In how many zeros does

25!

end?Divisibility & PrimesAMC 10 style~3m 6What is the smallest positive integer

x

x \equiv 2 \pmod 5

x \equiv 3 \pmod 7

?Modular ArithmeticAMC 10 style~4m 6What is the units digit of

1! + 2! + 3! + \cdots + 100!

?Modular ArithmeticAMC 10 style~3m 6How many ordered pairs of positive integers

(x, y)

xy + 3x + 2y = 30

?Diophantine EquationsAIME style~5m 6How many ordered pairs of positive integers

(x, y)

3x + 5y = 47

?Diophantine EquationsAMC 10 style~4m 6How many edges does the complete graph

K_8

have?Graph TheoryAMC 10 style~2m 6Define

f(1) = 1

f(n) = f(n-1) + n

n \ge 2

f(20)

?Recursive SequencesAMC 10 style~3m 7Let

S = \sum_{k=1}^{99} \tfrac{1}{k(k+1)}

S = \tfrac{m}{n}

in lowest terms, what is

m + n

?Sequences & SeriesAIME style~5m 7Ten people throw their hats in a pile and each picks one back uniformly at random. What is the expected number of people who get their own h…Expected ValueAIME style~4m 7A fair die is rolled repeatedly. What is the expected number of rolls until the first

6

appears?Expected ValueAMC 12 style~4m 7How many ordered pairs of positive integers

(x, y)

x^2 - y^2 = 45

?Diophantine EquationsAIME style~5m 7What is

\varphi(100)

, the number of integers in

[1,100]

100

?Euler Totient & CRTAMC 12 style~3m 7An

8 \times 8

board has two diagonally opposite corner squares removed. How many ways can the remaining

62

squares be tiled by

31

domi…Invariants & MonovariantsOlympiad classic~5m 7Signs

+

-

are placed between the numbers

1, 2, \ldots, 10

. What is the smallest nonnegative value the resulting expression can take…Invariants & MonovariantsOlympiad classic~5m 7What is the smallest

n

n

integers must contain two whose difference is divisible by

7

?Extremal PrincipleOlympiad classic~4m 7At a party of

9

people, is it possible that everyone shakes hands with exactly

3

others? Enter the number of such handshake configuratio…Graph TheoryOlympiad classic~4m 7A tree has

15

vertices. How many edges does it have?Graph TheoryOlympiad training~2m 7How many ways can a

2 \times 10

rectangle be tiled by

1 \times 2

dominoes?Recursive SequencesAIME style~5m 7A function satisfies

f(x + y) = f(x) + f(y)

x, y

f(1) = 3

f(7)

?Functional EquationsOlympiad training~3m 7What is the minimum value of

x + \tfrac{4}{x}

x > 0

?InequalitiesAMC 12 style~3m 7Quadrilateral

ABCD

\angle A = 70°

\angle C

in degrees?Advanced GeometryOlympiad training~2m 8What are the last two digits of

7^{222}

?Euler Totient & CRTAIME style~5m 8The numbers

1, 2, \ldots, 10

are on a board. A move erases two numbers

a, b

a + b + ab

. After nine moves one number remains…Invariants & MonovariantsOlympiad classic~8m 8A function satisfies

f(x) + 2f\!\left(\tfrac{1}{x}\right) = 3x

x \ne 0

f(2)

?Functional EquationsAMC 12 style~6m 8For positive reals

a, b

, what is the minimum value of

(a + b)\left(\tfrac{1}{a} + \tfrac{1}{b}\right)

?InequalitiesOlympiad training~4m 8How many ways can you make

10

1

2

5

-cent coins?Generating FunctionsOlympiad training~5m 9Equilateral triangle

ABC

is inscribed in a circle, and

P

BC

A

PB = 3

PC = 5

PA

?Advanced GeometryOlympiad classic~6m