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Showing 50 of 199 problems
1.
How many ordered pairs of integers (m,n)(m,n) satisfy n2−49=m\sqrt{n^2-49}=m?
(A)  1\text{(A)}\;1(B)  2\text{(B)}\;2(C)  3\text{(C)}\;3(D)  4\text{(D)}\;4(E)  infinitely many\text{(E)}\;\text{infinitely many}
2.
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 22 units. What is the volume of the new box?
(A)  245\text{(A)}\;\frac{24}{5}(B)  425\text{(B)}\;\frac{42}{5}(C)  815\text{(C)}\;\frac{81}{5}(D)  30\text{(D)}\;30(E)  48\text{(E)}\;48
3.
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\text{(A)}\;4(B)  6\text{(B)}\;6(C)  8\text{(C)}\;8(D)  12\text{(D)}\;12(E)  16\text{(E)}\;16
4.
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+2127\text{(A)}\;3^{127}+2^{127}(B)  3127+2127+2⋅363+3⋅263\text{(B)}\;3^{127}+2^{127}+2\cdot3^{63}+3\cdot2^{63}(C)  3128−2128\text{(C)}\;3^{128}-2^{128}(D)  3128+2128\text{(D)}\;3^{128}+2^{128}(E)  5127\text{(E)}\;5^{127}
5.
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\text{(A)}\;-88(B)  −80\text{(B)}\;-80(C)  −64\text{(C)}\;-64(D)  −41\text{(D)}\;-41(E)  −40\text{(E)}\;-40
6.
There exists a unique strictly increasing sequence of nonnegative integers a1<a2<⋯<aka_1<a_2<\cdots<a_k such that 2289+1217+1=2a1+2a2+⋯+2ak\frac{2^{289}+1}{2^{17}+1}=2^{a_1}+2^{a_2}+\cdots+2^{a_k}. What is kk?
(A)  117\text{(A)}\;117(B)  136\text{(B)}\;136(C)  137\text{(C)}\;137(D)  273\text{(D)}\;273(E)  306\text{(E)}\;306
7.
Real numbers xx and yy satisfy x+y=4x+y=4 and xy=−2xy=-2. What is the value of x+x3y2+y3x2+yx+\frac{x^3}{y^2}+\frac{y^3}{x^2}+y?
(A)  360\text{(A)}\;360(B)  400\text{(B)}\;400(C)  420\text{(C)}\;420(D)  440\text{(D)}\;440(E)  480\text{(E)}\;480
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)  9800\text{(A)}\;9800(B)  9900\text{(B)}\;9900(C)  10000\text{(C)}\;10000(D)  10100\text{(D)}\;10100(E)  10200\text{(E)}\;10200
9.
What is the sum of all real numbers xx for which ∣x2−12x+34∣=2\left|x^2-12x+34\right|=2?
(A)  12\text{(A)}\;12(B)  15\text{(B)}\;15(C)  18\text{(C)}\;18(D)  21\text{(D)}\;21(E)  25\text{(E)}\;25
10.
The sum of the first mm positive odd integers is 212212 more than the sum of the first nn positive even integers. What is the sum of all possible values of nn?
(A)  255\text{(A)}\;255(B)  256\text{(B)}\;256(C)  257\text{(C)}\;257(D)  258\text{(D)}\;258(E)  259\text{(E)}\;259
11.
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\text{(A)}\;\text{an infinite number}(B)  4\text{(B)}\;4(C)  2\text{(C)}\;2(D)  3\text{(D)}\;3(E)  0\text{(E)}\;0
12.
How many ordered pairs of integers (m,n)(m,n) satisfy m2+mn+n2=m2n2m^2+mn+n^2=m^2n^2?
(A)  7\text{(A)}\;7(B)  1\text{(B)}\;1(C)  3\text{(C)}\;3(D)  6\text{(D)}\;6(E)  5\text{(E)}\;5
13.
One of the following numbers is not divisible by any prime number less than 1010. Which is it?
(A)  2606−1\text{(A)}\;2^{606}-1(B)  2606+1\text{(B)}\;2^{606}+1(C)  2607−1\text{(C)}\;2^{607}-1(D)  2607+1\text{(D)}\;2^{607}+1(E)  2607+3607\text{(E)}\;2^{607}+3^{607}
14.
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\text{(A)}\;340(B)  360\text{(B)}\;360(C)  380\text{(C)}\;380(D)  400\text{(D)}\;400(E)  420\text{(E)}\;420
15.
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\text{(A)}\;2020(B)  2021\text{(B)}\;2021(C)  2022\text{(C)}\;2022(D)  2023\text{(D)}\;2023(E)  2024\text{(E)}\;2024
16.
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\text{(A)}\;10(B)  13\text{(B)}\;13(C)  19\text{(C)}\;19(D)  20\text{(D)}\;20(E)  23\text{(E)}\;23
17.
For each integer n≥2n\ge2, 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 1010 least values of nn such that SnS_n is divisible by 33?
(A)  196\text{(A)}\;196(B)  197\text{(B)}\;197(C)  198\text{(C)}\;198(D)  199\text{(D)}\;199(E)  200\text{(E)}\;200
18.
The real number xx satisfies the equation x+1x=5x+\frac{1}{x}=\sqrt5. What is the value of x11−7x7+x3x^{11}-7x^7+x^3?
(A)  −1\text{(A)}\;-1(B)  0\text{(B)}\;0(C)  1\text{(C)}\;1(D)  2\text{(D)}\;2(E)  5\text{(E)}\;\sqrt5
19.
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\text{(A)}\;2023(B)  2679\text{(B)}\;2679(C)  2941\text{(C)}\;2941(D)  3159\text{(D)}\;3159(E)  3235\text{(E)}\;3235
20.
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\text{(A)}\;12(B)  18\text{(B)}\;18(C)  24\text{(C)}\;24(D)  36\text{(D)}\;36(E)  42\text{(E)}\;42
21.
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\text{(A)}\;0(B)  1\text{(B)}\;1(C)  2\text{(C)}\;2(D)  3\text{(D)}\;3(E)  an infinite number\text{(E)}\;\text{an infinite number}
22.
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+1\text{(A)}\;x^2-x+1(B)  x2+x+1\text{(B)}\;x^2+x+1(C)  x4+1\text{(C)}\;x^4+1(D)  x6−x3+1\text{(D)}\;x^6-x^3+1(E)  x6+x3+1\text{(E)}\;x^6+x^3+1
23.
What is the number of terms with rational coefficients among the 10011001 terms in the expansion of (x23+y3)1000(x\sqrt[3]{2}+y\sqrt3)^{1000}?
(A)  0\text{(A)}\;0(B)  166\text{(B)}\;166(C)  167\text{(C)}\;167(D)  500\text{(D)}\;500(E)  501\text{(E)}\;501
24.
How many solutions does the equation sin⁡(π2cos⁡x)=cos⁡(π2sin⁡x)\sin\left(\frac{\pi}{2}\cos x\right)=\cos\left(\frac{\pi}{2}\sin x\right) have in the closed interval [0,π][0,\pi]?
(A)  0\text{(A)}\;0(B)  1\text{(B)}\;1(C)  2\text{(C)}\;2(D)  3\text{(D)}\;3(E)  4\text{(E)}\;4
25.
Let xx be the least real number greater than 11 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\text{(A)}\;10(B)  13\text{(B)}\;13(C)  14\text{(C)}\;14(D)  19\text{(D)}\;19(E)  20\text{(E)}\;20
26.
For how many ordered pairs (a,b)(a,b) of integers does the polynomial x3+ax2+bx+6x^3+ax^2+bx+6 have 33 distinct integer roots?
(A)  5\text{(A)}\;5(B)  6\text{(B)}\;6(C)  8\text{(C)}\;8(D)  7\text{(D)}\;7(E)  4\text{(E)}\;4
27.
For how many values of the constant kk will the polynomial x2+kx+36x^2+kx+36 have two distinct integer roots?
(A)  6\text{(A)}\;6(B)  8\text{(B)}\;8(C)  9\text{(C)}\;9(D)  14\text{(D)}\;14(E)  16\text{(E)}\;16
28.
Let c=2π11c=\frac{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\frac{\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\text{(A)}\;-1(B)  −115\text{(B)}\;-\frac{\sqrt{11}}{5}(C)  115\text{(C)}\;\frac{\sqrt{11}}{5}(D)  1011\text{(D)}\;\frac{10}{11}(E)  1\text{(E)}\;1
29.
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 aa, bb, and cc?
(A)  1+a+b+cc\text{(A)}\;\frac{1+a+b+c}{c}(B)  1+a+b+c\text{(B)}\;\text{1+a+b+c}(C)  1+a+b+cc2\text{(C)}\;\frac{1+a+b+c}{c^2}(D)  a+b+cc2\text{(D)}\;\frac{a+b+c}{c^2}(E)  1+a+b+ca+b+c\text{(E)}\;\frac{1+a+b+c}{a+b+c}
30.
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\text{(A)}\;\text{-z}(B)  −1\text{(B)}\;-1(C)  2021\text{(C)}\;2021(D)  z+1\text{(D)}\;\text{z+1}(E)  2z+1\text{(E)}\;\text{2z+1}
31.
What is the value of 13⋅24⋅35⋯1820⋅1921⋅2022\dfrac{1}{3}\cdot\dfrac{2}{4}\cdot\dfrac{3}{5}\cdots\dfrac{18}{20}\cdot\dfrac{19}{21}\cdot\dfrac{20}{22}?
(A)  1462\text{(A)}\;\frac{1}{462}(B)  1231\text{(B)}\;\frac{1}{231}(C)  1132\text{(C)}\;\frac{1}{132}(D)  2213\text{(D)}\;\frac{2}{213}(E)  122\text{(E)}\;\frac{1}{22}
32.
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.
33.
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.
34.
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)\geq a^2+b^2+c^2.
35.
Let 2n2^n be the greatest power of 22 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?
36.
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.
37.
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.
38.
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.
39.
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.
40.
What is the value of ∑n=2255log⁡2(1+1n)(log⁡2n)(log⁡2(n+1))?\sum_{n=2}^{255}\frac{\log_2\left(1+\frac1n\right)}{(\log_2 n)(\log_2(n+1))}?
(A)  34\text{(A)}\;\frac34(B)  1−1log⁡2255\text{(B)}\;1-\frac{1}{\log_2 255}(C)  78\text{(C)}\;\frac78(D)  1516\text{(D)}\;\frac{15}{16}(E)  1\text{(E)}\;1
41.
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!)^2y when it is written in its decimal representation.
42.
The harmonic mean of a collection of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers in the collection. For example, the harmonic mean of 4,4,4,4, and 55 is 113(14+14+15)=307\frac{1}{\frac13\left(\frac14+\frac14+\frac15\right)} = \frac{30}{7} What is the harmonic mean of all the real roots of the 4050th-degree polynomial ∏k=12025(kx2−4x−3)=(x2−4x−3)(2x2−4x−3)(3x2−4x−3)⋯(2025x2−4x−3)?\prod_{k=1}^{2025} \left(kx^2-4x-3\right) = (x^2-4x-3)(2x^2-4x-3)(3x^2-4x-3)\cdots(2025x^2-4x-3)?
(A)  −53\text{(A)}\;-\frac{5}{3}(B)  −32\text{(B)}\;-\frac{3}{2}(C)  −65\text{(C)}\;-\frac{6}{5}(D)  −56\text{(D)}\;-\frac{5}{6}(E)  −23\text{(E)}\;-\frac{2}{3}
43.
Triangle ABCABC has side lengths AB=80AB=80, BC=45BC=45, and AC=75AC=75. The bisector of ∠B\angle B and the altitude to side ABAB intersect at point PP. What is BPBP?
(A)  18\text{(A)}\;18(B)  19\text{(B)}\;19(C)  20\text{(C)}\;20(D)  21\text{(D)}\;21(E)  22\text{(E)}\;22
44.
Let aa, bb, and cc be the roots of the polynomial x3+kx+1x^3+kx+1. What is the sum a3b2+a2b3+b3c2+b2c3+c3a2+c2a3?a^3b^2+a^2b^3+b^3c^2+b^2c^3+c^3a^2+c^2a^3?
(A)  -k\text{(A)}\;\text{-k}(B)  -k+1\text{(B)}\;\text{-k+1}(C)  1\text{(C)}\;1(D)  k-1\text{(D)}\;\text{k-1}(E)  k\text{(E)}\;\text{k}
45.
The sum ∑k=1∞1k3+6k2+8k\sum_{k=1}^{\infty}\frac{1}{k^3+6k^2+8k} can be expressed as ab\frac{a}{b}, where aa and bb are relatively prime positive integers. What is a+ba+b?
(A)  89\text{(A)}\;89(B)  97\text{(B)}\;97(C)  102\text{(C)}\;102(D)  107\text{(D)}\;107(E)  129\text{(E)}\;129
46.
A frog hops along the number line according to the following rules.
  • It starts at 00.
  • If it is at 00, then it moves to 11 with probability 12\frac12 and it disappears with probability 12\frac12.
  • For n=1,2,n=1,2, or 33, if it is at nn, then it moves to n+1n+1 with probability 14\frac14, it moves to n−1n-1 with probability 14\frac14, and it disappears with probability 12\frac12.
What is the probability that the frog reaches 44?
(A)  1101\text{(A)}\;\frac{1}{101}(B)  1100\text{(B)}\;\frac{1}{100}(C)  199\text{(C)}\;\frac{1}{99}(D)  198\text{(D)}\;\frac{1}{98}(E)  197\text{(E)}\;\frac{1}{97}
47.
Awnik repeatedly plays a game that has a probability of winning of 13\frac13. The outcomes of the games are independent. What is the expected value of the number of games he will play until he has both won and lost at least once?
(A)  52\text{(A)}\;\frac52(B)  3\text{(B)}\;3(C)  165\text{(C)}\;\frac{16}{5}(D)  72\text{(D)}\;\frac72(E)  154\text{(E)}\;\frac{15}{4}
48.
Let SS denote the value of the infinite sum 19+199+1999+19999+⋯\frac19+\frac1{99}+\frac1{999}+\frac1{9999}+\cdots Find the remainder when the greatest integer less than or equal to 10100S10^{100}S is divided by 10001000.
49.
A standard fair six-sided die is rolled repeatedly. Each time the die reads 1 or 2, Alice gets a coin; each time it reads 3 or 4, Bob gets a coin; and each time it reads 5 or 6, Carol gets a coin. The probability that Alice and Bob each receive at least two coins before Carol receives any coins can be written as mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find 100m+n100m+n.
50.
Let NN be the number of positive integer divisors of 170171717017^{17} that leave a remainder of 55 upon division by 1212. Find the remainder when NN is divided by 10001000.