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AMC 10A 2024 (Problem 11)

How many ordered pairs of integers (m,n)(m, n) satisfy n2−49=m\sqrt{n^2 - 49} = m ?
(A) 1(B) 2(C) 3(D) 4(E) infinitely many

AMC 10/12A 2022 (Problem 16)

The roots of the polynomial 10x3−39x2+29x−610x^3 - 39x^2 + 29x - 6 are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box?
(A) 245\frac{24}{5}(B) 425\frac{42}{5}(C) 815\frac{81}{5}(D) 30(E) 48

AMC 10/12 A Fall 2021 (Problem 17)

For how many ordered pairs (b,c)(b,c) of positive integers does neither x2+bx+c=0x^2+bx+c=0 nor x2+cx+b=0x^2+cx+b=0 have two distinct real solutions?
(A) 4(B) 6(C) 8(D) 12(E) 16

AMC 10/12A Spring 2021 (Problem 10)

Which of the following is equivalent to (2+3)(22+32)(24+34)(28+38)(216+316)(232+332)(264+364)(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64}) ?
(A) 3127+21273^{127}+2^{127}(B) 3127+2127+2⋅363+3⋅2633^{127}+2^{127}+2\cdot3^{63}+3\cdot2^{63}(C) 3128−21283^{128}-2^{128}(D) 3128+21283^{128}+2^{128}(E) 51275^{127}

AMC 10/12A Spring 2021 (Problem 14)

All the roots of the polynomial z6−10z5+Az4+Bz3+Cz2+Dz+16z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16 are positive integers, possibly repeated. What is the value of BB?
(A) −88-88(B) −80-80(C) −64-64(D) −41-41(E) −40-40

AMC 10/12 A 2020 (Problem 21)

There exists a unique strictly increasing sequence of nonnegative integers a1<a2<⋯<aka_1 < a_2 < \dots < a_k such that 2289+1217+1=2a1+2a2+⋯+2ak\frac{2^{289} + 1}{2^{17} + 1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}. What is kk?
(A) 117(B) 136(C) 137(D) 273(E) 306

AMC 10A 2020 (Problem 14)

Real numbers xx and yy satisfy x+y=4x + y = 4 and x⋅y=−2x \cdot y = -2. What is the value of x+x3y2+y3x2+yx + \dfrac{x^3}{y^2} + \dfrac{y^3}{x^2} + y?
(A) 360360(B) 400400(C) 420420(D) 440440(E) 480480

AMC 10A 2020 (Problem 8)

What is the value of 1+2+3−4+5+6+7−8+⋯+197+198+199−2001 + 2 + 3 - 4 + 5 + 6 + 7 - 8 + \cdots + 197 + 198 + 199 - 200?
(A) 9,8009,800(B) 9,9009,900(C) 10,00010,000(D) 10,10010,100(E) 10,20010,200

AMC 10A 2020 (Problem 5)

What is the sum of all real numbers xx for which ∣x2−12x+34∣=2|x^2-12x+34|=2?
(A) 1212(B) 1515(C) 1818(D) 2121(E) 2525

AMC 10A 2012 (Problem 22)

The sum of the first mm positive odd integers is 212 more than the sum of the first nn positive even integers. What is the sum of all possible values of nn?
(A) 255(B) 256(C) 257(D) 258(E) 259

AMC 10B 2023 (Problem 22)

How many distinct values of xx satisfy ⌊x⌋2−3x+2=0\lfloor x \rfloor^2 - 3x + 2 = 0, where ⌊x⌋\lfloor x \rfloor denotes the greatest integer less than or equal to xx?
(A) an infinite number(B) 4(C) 2(D) 3(E) 0

AMC 10B 2023 (Problem 14)

How many ordered pairs of integers (m,n)(m,n) satisfy m2+mn+n2=m2n2m^2+mn+n^2=m^2n^2?
(A) 7(B) 1(C) 3(D) 6(E) 5

AMC 10B 2022 (Problem 17)

One of the following numbers is not divisible by any prime number less than 10. Which is it?
(A) 2606−12^{606}-1(B) 2606+12^{606}+1(C) 2607−12^{607}-1(D) 2607+12^{607}+1(E) 2607+36072^{607}+3^{607}

AMC 10B 2022 (Problem 15)

Let SnS_n be the sum of the first nn terms of an arithmetic sequence that has a common difference of 22. The quotient S3nSn\dfrac{S_{3n}}{S_n} does not depend on nn. What is S20S_{20}?
(A) 340(B) 360(C) 380(D) 400(E) 420

AMC 10B 2022 (Problem 9)

The sum 12!+23!+34!+⋯+20212022!\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \cdots + \frac{2021}{2022!} can be expressed as a−1b!a - \frac{1}{b!}, where aa and bb are positive integers. What is a+ba+b?
(A) 2020(B) 2021(C) 2022(D) 2023(E) 2024

AMC 10B 2022 (Problem 21)

Let P(x)P(x) be a polynomial with rational coefficients such that when P(x)P(x) is divided by x2+x+1x^2 + x + 1, the remainder is x+2x + 2, and when P(x)P(x) is divided by x2+1x^2 + 1, the remainder is 2x+12x + 1. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?
(A) 10(B) 13(C) 19(D) 20(E) 23

AMC 10B Fall 2021 (Problem 22)

For each integer n≥2n \ge 2, let SnS_n be the sum of all products jkjk, where jj and kk are integers and 1≤j<k≤n1 \le j < k \le n. What is the sum of the 10 least values of nn such that SnS_n is divisible by 3?
(A) 196(B) 197(C) 198(D) 199(E) 200

AMC 10B Spring 2021 (Problem 15)

The real number xx satisfies the equation x+1x=5x + \frac{1}{x} = \sqrt{5}. What is the value of x11−7x7+x3x^{11} - 7x^{7} + x^{3}?
(A) −1-1(B) 0(C) 1(D) 2(E) 5\sqrt{5}

AMC 12A 2023 (Problem 12)

What is the value of 23−13+43−33+63−53+⋯+183−1732^3-1^3+4^3-3^3+6^3-5^3+\cdots+18^3-17^3?
(A) 2023(B) 2679(C) 2941(D) 3159(E) 3235

AMC 12A 2023 (Problem 10)

Positive real numbers xx and yy satisfy y3=x2y^3=x^2 and (y−x)2=4y2(y-x)^2=4y^2. What is x+yx+y?
(A) 12(B) 18(C) 24(D) 36(E) 42

AMC 12A 2023 (Problem 23)

How many ordered pairs of positive real numbers (a,b)(a,b) satisfy the equation (1+2a)(2+2b)(2a+b)=32ab(1+2a)(2+2b)(2a+b)=32ab?
(A) 0(B) 1(C) 2(D) 3(E) an infinite number

AMC 12A 2022 (Problem 21)

Let P(x)=x2022+x1011+1P(x)=x^{2022}+x^{1011}+1. Which of the following polynomials is a factor of P(x)P(x)?
(A) x2−x+1x^2-x+1(B) x2+x+1x^2+x+1(C) x4+1x^4+1(D) x6−x3+1x^6-x^3+1(E) x6+x3+1x^6+x^3+1

AMC 12A Fall 2021 (Problem 12)

What is the number of terms with rational coefficients among the 1001 terms in the expansion of (x23+y3)1000(x\sqrt[3]{2}+y\sqrt{3})^{1000}?
(A) 0(B) 166(C) 167(D) 500(E) 501

AMC 12A Spring 2021 (Problem 19)

How many solutions does the equation sin⁡ ⁣(π2cos⁡x)=cos⁡ ⁣(π2sin⁡x)\sin\!\Big(\tfrac{\pi}{2}\cos x\Big) = \cos\!\Big(\tfrac{\pi}{2}\sin x\Big) have in the closed interval [0,π][0,\pi]?
(A) 0(B) 1(C) 2(D) 3(E) 4

AMC 12A Fall 2021 (Problem 19)

Let xx be the least real number greater than 1 such that sin⁡(x)=sin⁡(x2)\sin(x)=\sin(x^2), where the arguments are in degrees. What is xx rounded up to the closest integer?
(A) 10(B) 13(C) 14(D) 19(E) 20

AMC 12B 2023 (Problem 14)

For how many ordered pairs (a,b)(a,b) of integers does the polynomial x3+ax2+bx+6x^3 + ax^2 + bx + 6 have 3 distinct integer roots?
(A) 5(B) 6(C) 8(D) 7(E) 4

AMC 12B 2022 (Problem 4)

For how many values of the constant kk will the polynomial x2+kx+36x^2 + kx + 36 have two distinct integer roots?
(A) 6(B) 8(C) 9(D) 14(E) 16

AMC 12B 2021 Fall (Problem 13)

Let c=2π11c=\dfrac{2\pi}{11}. What is the value of sin⁡3c⋅sin⁡6c⋅sin⁡9c⋅sin⁡12c⋅sin⁡15csin⁡c⋅sin⁡2c⋅sin⁡3c⋅sin⁡4c⋅sin⁡5c\dfrac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?
(A) −1-1(B) −115-\dfrac{\sqrt{11}}{5}(C) 115\dfrac{\sqrt{11}}{5}(D) 1011\dfrac{10}{11}(E) 1

AMC 12B 2021 Spring (Problem 16)

Let g(x)g(x) be a polynomial with leading coefficient 11, whose three roots are the reciprocals of the three roots of f(x)=x3+ax2+bx+cf(x) = x^3 + ax^2 + bx + c, where 1<a<b<c1 < a < b < c. What is g(1)g(1) in terms of a,b,a,b, and cc?
(A) 1+a+b+cc\dfrac{1 + a + b + c}{c}(B) 1+a+b+c1 + a + b + c(C) 1+a+b+cc2\dfrac{1 + a + b + c}{c^2}(D) a+b+cc2\dfrac{a + b + c}{c^2}(E) 1+a+b+ca+b+c\dfrac{1 + a + b + c}{a + b + c}

AMC 12B 2021 Spring (Problem 20)

Let Q(z)Q(z) and R(z)R(z) be the unique polynomials such that z2021+1=(z2+z+1) Q(z)+R(z)z^{2021}+1=(z^2+z+1)\,Q(z)+R(z) and the degree of RR is less than 22. What is R(z)R(z)?
(A) −z-z(B) −1-1(C) 2021(D) z+1z+1(E) 2z+12z+1

AMC 8 2022 (Problem 8)

What is the value of 13⋅24⋅35⋯1820⋅1921⋅2022\frac{1}{3}\cdot\frac{2}{4}\cdot\frac{3}{5}\cdots\frac{18}{20}\cdot\frac{19}{21}\cdot\frac{20}{22}?
(A) 1462\frac{1}{462}(B) 1231\frac{1}{231}(C) 1132\frac{1}{132}(D) 2213\frac{2}{213}(E) 122\frac{1}{22}

AIME II 2022 (Problem 10)

Find the remainder when ((32)2)+((42)2)+⋯+((402)2)\binom{\binom{3}{2}}{2}+\binom{\binom{4}{2}}{2}+\cdots+\binom{\binom{40}{2}}{2} is divided by 10001000.

USAJMO 2023 (Problem 1)

Find all triples of positive integers (x,y,z)(x, y, z) that satisfy the equation 2(x+y+z+2xyz)2=(2xy+2yz+2zx+1)2+20232(x + y + z + 2xyz)^2 = (2xy + 2yz + 2zx + 1)^2 + 2023.

USAMO 2018 (Problem 1)

Let a,b,ca, b, c be positive real numbers such that a+b+c=4abc3a + b + c = 4\sqrt[3]{abc}. Prove that 2(ab+bc+ca)+4min⁡(a2,b2,c2)≥a2+b2+c22(ab + bc + ca) + 4\min(a^2, b^2, c^2) \ge a^2 + b^2 + c^2.

MathCounts 2025 Chapter — Sprint Round (Problem 27)

Let 2n2^n be the greatest power of 2 that divides 1×2×3×4+2×3×4×5+3×4×5×6+⋯+25×26×27×281\times2\times3\times4 + 2\times3\times4\times5 + 3\times4\times5\times6 + \cdots + 25\times26\times27\times28. What is the value of nn?

MathCounts 2025 State — Sprint Round (Problem 30)

If xx and yy are real numbers such that (4−x)(4+y)=2(4 - x)(4 + y) = 2 and (4+x)(4−y)=3(4 + x)(4 - y) = 3, what is the value of (x2−1)(y2−1)(x^2 - 1)(y^2 - 1)? Express your answer as a common fraction.

HMMT 2025 — Algebra & Number Theory Round (Problem 7)

There exists a unique triple (a,b,c)(a, b, c) of positive real numbers that satisfies the equations 2(a2+1)=3(b2+1)=4(c2+1)2(a^2 + 1) = 3(b^2 + 1) = 4(c^2 + 1) and ab+bc+ca=1ab + bc + ca = 1. Compute a+b+ca + b + c.

BMO1 2016/2017 (Problem 3)

Determine all pairs (m,n)(m,n) of positive integers which satisfy the equation n2−6n=m2+m−10n^2 - 6n = m^2 + m - 10.

SMO 2025 Junior (Problem 8)

If xx and yy are positive integers such that xy−9x−9y=20xy - 9x - 9y = 20, find the value of x2+y2x^2 + y^2.

SMO 2025 Senior (Problem 20)

The roots of the polynomial 10x3−39x2+29x−610x^3 - 39x^2 + 29x - 6 are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box?
(A) 245\dfrac{24}{5}(B) 425\dfrac{42}{5}(C) 815\dfrac{81}{5}(D) 30(E) 48

SMO 2025 Open (Problem 7)

Let y=∑k=020(20k)2y = \sum_{k=0}^{20} \binom{20}{k}^2. Find the number of consecutive zeros at the end of the number (20!)2y(20!)^2 y when it is written in its decimal representation.