query_id stringlengths 22 42 | query stringlengths 85 1.17k | positive_passages listlengths 1 8 | negative_passages listlengths 0 0 | hard_negative_passages listlengths 0 0 |
|---|---|---|---|---|
aops_2019_AMC_12A_Problems/Problem_17 | Let $s_k$ denote the sum of the $\textit{k}$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \, s_k + b \, s_{k-1} + c \, s_{k-2}$ for $k = 2$, $3$, $....$ What is $a+b+c$?
$\textbf{(A)} \; -6 \qquad \textbf{(B)} \; 0 \qquad \textbf{(C)} \; 6 \qquad \textbf{(D)} \; 10 \qquad \textbf{(E)} \; 26$ | [
{
"docid": "math_test_intermediate_algebra_1179",
"text": "Consider the polynomials $P(x) = x^6-x^5-x^3-x^2-x$ and $Q(x)=x^4-x^3-x^2-1$. Given that $z_1, z_2, z_3$, and $z_4$ are the roots of $Q(x)=0$, find $P(z_1)+P(z_2)+P(z_3)+P(z_4).$\nWe perform polynomial division with $P(x)$ as the dividend and $Q(x)... | [] | [] |
math_test_intermediate_algebra_1179 | Consider the polynomials $P(x) = x^{6} - x^{5} - x^{3} - x^{2} - x$ and $Q(x) = x^{4} - x^{3} - x^{2} - 1.$ Given that $z_{1},z_{2},z_{3},$ and $z_{4}$ are the roots of $Q(x) = 0,$ find $P(z_{1}) + P(z_{2}) + P(z_{3}) + P(z_{4}).$ | [
{
"docid": "aops_2019_AMC_12A_Problems/Problem_17",
"text": "Let $s_k$ denote the sum of the $\\textit{k}$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \\, s_k + b \\, s_{k-1} + c \\, s_{k... | [] | [] |
math_train_intermediate_algebra_513 | Let $r$, $s$, and $t$ be the three roots of the equation
$8x^3 + 1001x + 2008 = 0.$
Find $(r + s)^3 + (s + t)^3 + (t + r)^3$. | [
{
"docid": "aops_2019_AMC_12A_Problems/Problem_17",
"text": "Let $s_k$ denote the sum of the $\\textit{k}$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \\, s_k + b \\, s_{k-1} + c \\, s_{k... | [] | [] |
aops_2015_AMC_10B_Problems/Problem_15 | The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?
$\textbf{(A) }41\qquad\textbf{(B) }47\qquad\textbf{(C) }59\qquad\textbf{(D) }61\qquad\textbf{(E) }66$ | [
{
"docid": "math_train_counting_and_probability_5024",
"text": "Ninety-four bricks, each measuring $4''\\times10''\\times19'',$ are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''\\,$ or $10''\\,$ or $19''\\,$ to the total height of the t... | [] | [] |
math_train_counting_and_probability_5024 | Ninety-four bricks, each measuring $4''\times10''\times19'',$ are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''\,$ or $10''\,$ or $19''\,$ to the total height of the tower. How many different tower heights can be achieved using all ninety-four of the bricks? | [
{
"docid": "aops_2015_AMC_10B_Problems/Problem_15",
"text": "The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?\n$\\textbf{(A) }41\\qquad\\... | [] | [] |
math_train_number_theory_7095 | Find the sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $5,n,$ and $n+1$ cents, $91$ cents is the greatest postage that cannot be formed. | [
{
"docid": "aops_2015_AMC_10B_Problems/Problem_15",
"text": "The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?\n$\\textbf{(A) }41\\qquad\\... | [] | [] |
math_train_number_theory_839 | Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$
\text A. \ \ 21 \qquad \text B. \ \ 60 \qquad \text C. \ \ 119 \qquad \text D. \ \ 180 \qquad \text E. \ \ 231
$$ | [
{
"docid": "math_train_number_theory_7012",
"text": "Find $3x^2 y^2$ if $x$ and $y$ are integers such that $y^2 + 3x^2 y^2 = 30x^2 + 517$.\n\nIf we move the $x^2$ term to the left side, it is factorable:\n\\[(3x^2 + 1)(y^2 - 10) = 517 - 10\\]\n$507$ is equal to $3 \\cdot 13^2$. Since $x$ and $y$ are integer... | [] | [] |
math_train_number_theory_7012 | Find $3x^2 y^2$ if $x$ and $y$ are integers such that $y^2 + 3x^2 y^2 = 30x^2 + 517$. | [
{
"docid": "math_train_number_theory_839",
"text": "Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$\n\\text A. \\ \\ 21 \\qquad \\text B. \\ \\ 60 \\qquad \\text C. \\ \\ 119 \\qquad \\text D. ... | [] | [] |
aops_2008_AMC_12B_Problems/Problem_16 | A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $(a,b)$?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$ | [
{
"docid": "math_train_number_theory_839",
"text": "Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$\n\\text A. \\ \\ 21 \\qquad \\text B. \\ \\ 60 \\qquad \\text C. \\ \\ 119 \\qquad \\text D. ... | [] | [] |
aops_2000_AIME_I_Problems/Problem_9 | The system of equations
\begin{eqnarray*}\log_{10}(2000xy) - (\log_{10}x)(\log_{10}y) & = & 4 \\ \log_{10}(2yz) - (\log_{10}y)(\log_{10}z) & = & 1 \\ \log_{10}(zx) - (\log_{10}z)(\log_{10}x) & = & 0 \\ \end{eqnarray*}
has two solutions $(x_{1},y_{1},z_{1})$ and $(x_{2},y_{2},z_{2})$. Find $y_{1} + y_{2}$. | [
{
"docid": "math_train_number_theory_839",
"text": "Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$\n\\text A. \\ \\ 21 \\qquad \\text B. \\ \\ 60 \\qquad \\text C. \\ \\ 119 \\qquad \\text D. ... | [] | [] |
math_train_number_theory_7030 | The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}$? | [
{
"docid": "math_train_number_theory_839",
"text": "Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$\n\\text A. \\ \\ 21 \\qquad \\text B. \\ \\ 60 \\qquad \\text C. \\ \\ 119 \\qquad \\text D. ... | [] | [] |
aops_2008_AMC_12A_Problems/Problem_15 | Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?
$\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ 6\qquad\mathrm{(E)}\ 8$ | [
{
"docid": "math_train_number_theory_7016",
"text": "One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that $133^5+110^5+84^5+27^5=n^{5}$. Find the value of $n$.\n\nNote that $n$ is even, since the $LHS$ consists of two ... | [] | [] |
math_train_number_theory_7016 | One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that $133^5+110^5+84^5+27^5=n^{5}.$ Find the value of $n$. | [
{
"docid": "aops_2008_AMC_12A_Problems/Problem_15",
"text": "Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?\n$\\mathrm{(A)}\\ 0\\qquad\\mathrm{(B)}\\ 2\\qquad\\mathrm{(C)}\\ 4\\qquad\\mathrm{(D)}\\ 6\\qquad\\mathrm{(E)}\\ 8$\n$k \\equiv 2008^2 + 2^{2008} \\equiv 8^2 + 2^4 \\equiv 4+6 \... | [] | [] |
aops_2021_AIME_I_Problems/Problem_14 | For any positive integer $a, \sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a$. Find the sum of the prime factors in the prime factorization of $n$. | [
{
"docid": "aops_2008_AMC_12A_Problems/Problem_15",
"text": "Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?\n$\\mathrm{(A)}\\ 0\\qquad\\mathrm{(B)}\\ 2\\qquad\\mathrm{(C)}\\ 4\\qquad\\mathrm{(D)}\\ 6\\qquad\\mathrm{(E)}\\ 8$\n$k \\equiv 2008^2 + 2^{2008} \\equiv 8^2 + 2^4 \\equiv 4+6 \... | [] | [] |
aops_2005_IMO_Problems/Problem_4 | Determine all positive integers relatively prime to all the terms of the infinite sequence $a_n=2^n+3^n+6^n -1,\ n\geq 1.$ | [
{
"docid": "aops_2008_AMC_12A_Problems/Problem_15",
"text": "Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?\n$\\mathrm{(A)}\\ 0\\qquad\\mathrm{(B)}\\ 2\\qquad\\mathrm{(C)}\\ 4\\qquad\\mathrm{(D)}\\ 6\\qquad\\mathrm{(E)}\\ 8$\n$k \\equiv 2008^2 + 2^{2008} \\equiv 8^2 + 2^4 \\equiv 4+6 \... | [] | [] |
aops_2024_AIME_I_Problems/Problem_13 | Let $p$ be the least prime number for which there exists a positive integer $n$ such that $n^{4}+1$ is divisible by $p^{2}$. Find the least positive integer $m$ such that $m^{4}+1$ is divisible by $p^{2}$. | [
{
"docid": "aops_2008_AMC_12A_Problems/Problem_15",
"text": "Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?\n$\\mathrm{(A)}\\ 0\\qquad\\mathrm{(B)}\\ 2\\qquad\\mathrm{(C)}\\ 4\\qquad\\mathrm{(D)}\\ 6\\qquad\\mathrm{(E)}\\ 8$\n$k \\equiv 2008^2 + 2^{2008} \\equiv 8^2 + 2^4 \\equiv 4+6 \... | [] | [] |
aops_2017_AMC_10B_Problems/Problem_14 | An integer $N$ is selected at random in the range $1\leq N \leq 2020$ . What is the probability that the remainder when $N^{16}$ is divided by $5$ is $1$?
$\textbf{(A)}\ \frac{1}{5}\qquad\textbf{(B)}\ \frac{2}{5}\qquad\textbf{(C)}\ \frac{3}{5}\qquad\textbf{(D)}\ \frac{4}{5}\qquad\textbf{(E)}\ 1$ | [
{
"docid": "aops_2008_AMC_12A_Problems/Problem_15",
"text": "Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?\n$\\mathrm{(A)}\\ 0\\qquad\\mathrm{(B)}\\ 2\\qquad\\mathrm{(C)}\\ 4\\qquad\\mathrm{(D)}\\ 6\\qquad\\mathrm{(E)}\\ 8$\n$k \\equiv 2008^2 + 2^{2008} \\equiv 8^2 + 2^4 \\equiv 4+6 \... | [] | [] |
aops_2005_USAMO_Problems/Problem_2 | Prove that the system
\begin{align*}x^6 + x^3 + x^3y + y &= 147^{157} \\ x^3 + x^3y + y^2 + y + z^9 &= 157^{147}\end{align*}
has no solutions in integers $x$, $y$, and $z$. | [
{
"docid": "aops_2008_AMC_12A_Problems/Problem_15",
"text": "Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?\n$\\mathrm{(A)}\\ 0\\qquad\\mathrm{(B)}\\ 2\\qquad\\mathrm{(C)}\\ 4\\qquad\\mathrm{(D)}\\ 6\\qquad\\mathrm{(E)}\\ 8$\n$k \\equiv 2008^2 + 2^{2008} \\equiv 8^2 + 2^4 \\equiv 4+6 \... | [] | [] |
math_train_counting_and_probability_5035 | Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the cafeteria is $40 \%,$ and $m = a - b\sqrt {c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a + b + c.$ | [
{
"docid": "aops_2004_AIME_I_Problems/Problem_10",
"text": "A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n,$ where $m$ and $n$ are relatively prime... | [] | [] |
aops_2004_AIME_I_Problems/Problem_10 | A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ | [
{
"docid": "math_train_counting_and_probability_5035",
"text": "Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the... | [] | [] |
math_train_geometry_6173 | Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\dfrac{1}{2}$ is $\dfrac{a-b\pi}{c}$, where $a$, $b$, and $c$ are positive integers with $\gcd(a,b,c)=1$. What is $a+b+c$?
$\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$ | [
{
"docid": "math_train_counting_and_probability_5035",
"text": "Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while the other is in the... | [] | [] |
math_train_geometry_6018 | A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A_{}^{}$. | [
{
"docid": "aops_2023_AIME_I_Problems/Problem_5",
"text": "Let $P$ be a point on the circle circumscribing square $ABCD$ that satisfies $PA \\cdot PC = 56$ and $PB \\cdot PD = 90.$ Find the area of $ABCD.$\nPtolemy's theorem states that for cyclic quadrilateral $WXYZ$, $WX\\cdot YZ + XY\\cdot WZ = WY\\cdot ... | [] | [] |
aops_2023_AIME_I_Problems/Problem_5 | Let $P$ be a point on the circle circumscribing square $ABCD$ that satisfies $PA \cdot PC = 56$ and $PB \cdot PD = 90.$ Find the area of $ABCD.$ | [
{
"docid": "math_train_geometry_6018",
"text": "A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A$.\n\n[asy]defaultpen(fontsize(9)); pair A=expi(-pi/... | [] | [] |
math_test_geometry_454 | In triangle $ABC$ we have $AB=7$, $AC=8$, $BC=9$. Point $D$ is on the circumscribed circle of the triangle so that $AD$ bisects angle $BAC$. What is the value of $\frac{AD}{CD}$?
$\text{(A) } \dfrac{9}{8} \qquad \text{(B) } \dfrac{5}{3} \qquad \text{(C) } 2 \qquad \text{(D) } \dfrac{17}{7} \qquad \text{(E) } \dfrac{5}{2}$ | [
{
"docid": "math_train_geometry_6018",
"text": "A hexagon is inscribed in a circle. Five of the sides have length $81$ and the sixth, denoted by $\\overline{AB}$, has length $31$. Find the sum of the lengths of the three diagonals that can be drawn from $A$.\n\n[asy]defaultpen(fontsize(9)); pair A=expi(-pi/... | [] | [] |
math_train_counting_and_probability_5011 | In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence TTTHHTHTTTHHTTH of 15 coin tosses we observe that there are two HH, three HT, four TH, and five TT subsequences. How many different sequences of 15 coin tosses will contain exactly two HH, three HT, four TH, and five TT subsequences? | [
{
"docid": "aops_2001_AMC_10_Problems/Problem_19",
"text": "Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?\n$\\textbf{(A)}\\ 6 \\qquad \\textbf{(B)}\\ 9 \\qquad \\textbf{(C)}\\ 12 \\qquad \\textbf{(D)}\... | [] | [] |
aops_2001_AMC_10_Problems/Problem_19 | Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 18$ | [
{
"docid": "math_train_counting_and_probability_5011",
"text": "In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence TTTHHTHTTTHH... | [] | [] |
math_test_counting_and_probability_1009 | Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?
$\mathrm{(A) \ } 22\qquad \mathrm{(B) \ } 25\qquad \mathrm{(C) \ } 27\qquad \mathrm{(D) \ } 28\qquad \mathrm{(E) \ } 729$ | [
{
"docid": "math_train_counting_and_probability_5011",
"text": "In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence TTTHHTHTTTHH... | [] | [] |
math_train_counting_and_probability_5131 | Let $N$ denote the number of $7$ digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when $N$ is divided by $1000$. (Repeated digits are allowed.) | [
{
"docid": "math_train_counting_and_probability_5011",
"text": "In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence TTTHHTHTTTHH... | [] | [] |
aops_2007_AIME_I_Problems/Problem_10 | In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let $N$ be the number of shadings with this property. Find the remainder when $N$ is divided by 1000.
| [
{
"docid": "math_train_counting_and_probability_5011",
"text": "In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence TTTHHTHTTTHH... | [] | [] |
aops_2018_AMC_8_Problems/Problem_23 | From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?
[asy] size(3cm); pair A[]; for (int i=0; i<9; ++i) { A[i] = rotate(22.5+45*i)*(1,0); } filldraw(A[0]--A[1]--A[2]--A[3]--A[4]--A[5]--A[6]--A[7]--cycle,gray,black); for (int i=0; i<8; ++i) { dot(A[i]); } [/asy]
$\textbf{(A) } \frac{2}{7} \qquad \textbf{(B) } \frac{5}{42} \qquad \textbf{(C) } \frac{11}{14} \qquad \textbf{(D) } \frac{5}{7} \qquad \textbf{(E) } \frac{6}{7}$ | [
{
"docid": "math_train_counting_and_probability_5011",
"text": "In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence TTTHHTHTTTHH... | [] | [] |
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