question stringclasses 100
values | answer stringclasses 92
values | metadata dict |
|---|---|---|
Let $T=8$. At a party, everyone shakes hands with everyone else exactly once, except Ed, who leaves early. A grand total of $20 T$ handshakes take place. Compute the number of people at the party who shook hands with Ed. | 7 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2612,
"problem_type": "Combinatorics",
"unit": null
} |
Let $T$ = 2069, and let $K$ be the sum of the digits of $T$. Let $r$ and $s$ be the two roots of the polynomial $x^{2}-18 x+K$. Compute $|r-s|$. | 16 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2773,
"problem_type": "Algebra",
"unit": null
} |
Let $x$ be the smallest positive integer such that $1584 \cdot x$ is a perfect cube, and let $y$ be the smallest positive integer such that $x y$ is a multiple of 1584 . Compute $y$. | 12 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2708,
"problem_type": "Number Theory",
"unit": null
} |
Determine all values of $x$ for which $2^{\log _{10}\left(x^{2}\right)}=3\left(2^{1+\log _{10} x}\right)+16$. | 1000 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2256,
"problem_type": "Algebra",
"unit": null
} |
Let $T=17$. In ARMLovia, the unit of currency is the edwah. Janet's wallet contains bills in denominations of 20 and 80 edwahs. If the bills are worth an average of $2 T$ edwahs each, compute the smallest possible value of the bills in Janet's wallet. | 1020 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 3049,
"problem_type": "Combinatorics",
"unit": null
} |
Let $W=(0,0), A=(7,0), S=(7,1)$, and $H=(0,1)$. Compute the number of ways to tile rectangle $W A S H$ with triangles of area $1 / 2$ and vertices at lattice points on the boundary of WASH. | 3432 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2643,
"problem_type": "Geometry",
"unit": null
} |
Let $T=67$. Rectangles $F A K E$ and $F U N K$ lie in the same plane. Given that $E F=T$, $A F=\frac{4 T}{3}$, and $U F=\frac{12}{5}$, compute the area of the intersection of the two rectangles. | 262 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2990,
"problem_type": "Geometry",
"unit": null
} |
A fair coin is flipped $n$ times. Compute the smallest positive integer $n$ for which the probability that the coin has the same result every time is less than $10 \%$. | 5 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2925,
"problem_type": "Combinatorics",
"unit": null
} |
Elizabeth is in an "escape room" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to "flip" a switch means to turn it on if it is currently off, and of... | 76 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 3000,
"problem_type": "Combinatorics",
"unit": null
} |
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}_{>0} \rightarrow \mathbb{Q}_{>0}$ satisfying
$$
f\left(x^{2} f(y)^{2}\right)=f(x)^{2} f(y)
\tag{*}
$$
for all $x, y \in \mathbb{Q}_{>0}$. | f(x)=1 | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2029,
"problem_type": "Algebra",
"unit": null
} |
Let $T=36$. Square $A B C D$ has area $T$. Points $M, N, O$, and $P$ lie on $\overline{A B}$, $\overline{B C}, \overline{C D}$, and $\overline{D A}$, respectively, so that quadrilateral $M N O P$ is a rectangle with $M P=2$. Compute $M N$. | 6 \sqrt{2}-2 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2745,
"problem_type": "Geometry",
"unit": null
} |
Let $T=$ 21. The number $20^{T} \cdot 23^{T}$ has $K$ positive divisors. Compute the greatest prime factor of $K$. | 43 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2776,
"problem_type": "Number Theory",
"unit": null
} |
Alice fills the fields of an $n \times n$ board with numbers from 1 to $n^{2}$, each number being used exactly once. She then counts the total number of good paths on the board. A good path is a sequence of fields of arbitrary length (including 1) such that:
(i) The first field in the sequence is one that is only adja... | 2 n^{2}-2 n+1 | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2011,
"problem_type": "Combinatorics",
"unit": null
} |
Determine the number of quadruples of positive integers $(a, b, c, d)$ with $a<b<c<d$ that satisfy both of the following system of equations:
$$
\begin{aligned}
a c+a d+b c+b d & =2023 \\
a+b+c+d & =296
\end{aligned}
$$ | 417 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2408,
"problem_type": "Number Theory",
"unit": null
} |
$\quad$ Compute the greatest real number $K$ for which the graphs of
$$
(|x|-5)^{2}+(|y|-5)^{2}=K \quad \text { and } \quad(x-1)^{2}+(y+1)^{2}=37
$$
have exactly two intersection points. | 29 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 3016,
"problem_type": "Geometry",
"unit": null
} |
Find the largest possible integer $k$, such that the following statement is true:
Let 2009 arbitrary non-degenerated triangles be given. In every triangle the three sides are colored, such that one is blue, one is red and one is white. Now, for every color separately, let us sort the lengths of the sides. We obtain
$... | 1 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2082,
"problem_type": "Algebra",
"unit": null
} |
On a flat plane in Camelot, King Arthur builds a labyrinth $\mathfrak{L}$ consisting of $n$ walls, each of which is an infinite straight line. No two walls are parallel, and no three walls have a common point. Merlin then paints one side of each wall entirely red and the other side entirely blue.
At the intersection o... | k=n+1 | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 1915,
"problem_type": "Combinatorics",
"unit": null
} |
The string $A A A B B B A A B B$ is a string of ten letters, each of which is $A$ or $B$, that does not include the consecutive letters $A B B A$.
The string $A A A B B A A A B B$ is a string of ten letters, each of which is $A$ or $B$, that does include the consecutive letters $A B B A$.
Determine, with justificatio... | 631 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2525,
"problem_type": "Combinatorics",
"unit": null
} |
Elizabeth is in an "escape room" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to "flip" a switch means to turn it on if it is currently off, and of... | \frac{n}{2} | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2995,
"problem_type": "Combinatorics",
"unit": null
} |
If $n$ is a positive integer, then $n$ !! is defined to be $n(n-2)(n-4) \cdots 2$ if $n$ is even and $n(n-2)(n-4) \cdots 1$ if $n$ is odd. For example, $8 ! !=8 \cdot 6 \cdot 4 \cdot 2=384$ and $9 ! !=9 \cdot 7 \cdot 5 \cdot 3 \cdot 1=945$. Compute the number of positive integers $n$ such that $n !$ ! divides 2012!!. | 1510 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2793,
"problem_type": "Number Theory",
"unit": null
} |
A permutation of a list of numbers is an ordered arrangement of the numbers in that list. For example, $3,2,4,1,6,5$ is a permutation of $1,2,3,4,5,6$. We can write this permutation as $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$, where $a_{1}=3, a_{2}=2, a_{3}=4, a_{4}=1, a_{5}=6$, and $a_{6}=5$.
Determine the average v... | \frac{10}{3} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2430,
"problem_type": "Combinatorics",
"unit": null
} |
Six solid regular tetrahedra are placed on a flat surface so that their bases form a regular hexagon $\mathcal{H}$ with side length 1 , and so that the vertices not lying in the plane of $\mathcal{H}$ (the "top" vertices) are themselves coplanar. A spherical ball of radius $r$ is placed so that its center is directly a... | \frac{\sqrt{2}}{3} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2956,
"problem_type": "Geometry",
"unit": null
} |
$A \pm 1 \text{-}sequence$ is a sequence of 2022 numbers $a_{1}, \ldots, a_{2022}$, each equal to either +1 or -1 . Determine the largest $C$ so that, for any $\pm 1 -sequence$, there exists an integer $k$ and indices $1 \leqslant t_{1}<\ldots<t_{k} \leqslant 2022$ so that $t_{i+1}-t_{i} \leqslant 2$ for all $i$, and
... | 506 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2004,
"problem_type": "Combinatorics",
"unit": null
} |
Determine all real solutions to the system of equations
$$
\begin{aligned}
& x+\log _{10} x=y-1 \\
& y+\log _{10}(y-1)=z-1 \\
& z+\log _{10}(z-2)=x+2
\end{aligned}
$$
and prove that there are no more solutions. | 1,2,3 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2575,
"problem_type": "Algebra",
"unit": null
} |
Parallelogram $A B C D$ is rotated about $A$ in the plane, resulting in $A B^{\prime} C^{\prime} D^{\prime}$, with $D$ on $\overline{A B^{\prime}}$. Suppose that $\left[B^{\prime} C D\right]=\left[A B D^{\prime}\right]=\left[B C C^{\prime}\right]$. Compute $\tan \angle A B D$. | \sqrt{2}-1$,$\frac{3-\sqrt{2}}{7} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2771,
"problem_type": "Geometry",
"unit": null
} |
$\quad$ Let $T=40$. If $x+9 y=17$ and $T x+(T+1) y=T+2$, compute $20 x+14 y$. | 8 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2651,
"problem_type": "Algebra",
"unit": null
} |
Compute all ordered pairs of real numbers $(x, y)$ that satisfy both of the equations:
$$
x^{2}+y^{2}=6 y-4 x+12 \quad \text { and } \quad 4 y=x^{2}+4 x+12
$$ | (-6,6)$, $(2,6) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2950,
"problem_type": "Algebra",
"unit": null
} |
Suppose that the functions $f(x)$ and $g(x)$ satisfy the system of equations
$$
\begin{aligned}
f(x)+3 g(x) & =x^{2}+x+6 \\
2 f(x)+4 g(x) & =2 x^{2}+4
\end{aligned}
$$
for all $x$. Determine the values of $x$ for which $f(x)=g(x)$. | 5,-2 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2439,
"problem_type": "Algebra",
"unit": null
} |
A plane has a special point $O$ called the origin. Let $P$ be a set of 2021 points in the plane, such that
(i) no three points in $P$ lie on a line and
(ii) no two points in $P$ lie on a line through the origin.
A triangle with vertices in $P$ is $f a t$, if $O$ is strictly inside the triangle. Find the maximum numb... | 2021 \cdot 505 \cdot 337 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2174,
"problem_type": "Geometry",
"unit": null
} |
Polygon $A_{1} A_{2} \ldots A_{n}$ is a regular $n$-gon. For some integer $k<n$, quadrilateral $A_{1} A_{2} A_{k} A_{k+1}$ is a rectangle of area 6 . If the area of $A_{1} A_{2} \ldots A_{n}$ is 60 , compute $n$. | 40 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 3037,
"problem_type": "Geometry",
"unit": null
} |
Consider the following system of equations in which all logarithms have base 10:
$$
\begin{aligned}
(\log x)(\log y)-3 \log 5 y-\log 8 x & =a \\
(\log y)(\log z)-4 \log 5 y-\log 16 z & =b \\
(\log z)(\log x)-4 \log 8 x-3 \log 625 z & =c
\end{aligned}
$$
If $a=-4, b=4$, and $c=-18$, solve the system of equations. | (10^{4}, 10^{3}, 10^{10}),(10^{2}, 10^{-1}, 10^{-2}) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2303,
"problem_type": "Algebra",
"unit": null
} |
Let $T=60$ . Lydia is a professional swimmer and can swim one-fifth of a lap of a pool in an impressive 20.19 seconds, and she swims at a constant rate. Rounded to the nearest integer, compute the number of minutes required for Lydia to swim $T$ laps. | 101 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2986,
"problem_type": "Algebra",
"unit": null
} |
Kerry has a list of $n$ integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying $a_{1} \leq a_{2} \leq \ldots \leq a_{n}$. Kerry calculates the pairwise sums of all $m=\frac{1}{2} n(n-1)$ possible pairs of integers in her list and orders these pairwise sums as $s_{1} \leq s_{2} \leq \ldots \leq s_{m}$. For example, if Kerry'... | (1,7,103, 105), (3, 5, 101, 107) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2336,
"problem_type": "Number Theory",
"unit": null
} |
Eleanor has 100 marbles, each of which is black or gold. The ratio of the number of black marbles to the number of gold marbles is $1: 4$. How many gold marbles should she add to change this ratio to $1: 6$ ? | 40 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2322,
"problem_type": "Combinatorics",
"unit": null
} |
Let $T=\frac{9}{17}$. When $T$ is expressed as a reduced fraction, let $m$ and $n$ be the numerator and denominator, respectively. A square pyramid has base $A B C D$, the distance from vertex $P$ to the base is $n-m$, and $P A=P B=P C=P D=n$. Compute the area of square $A B C D$. | 450 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2838,
"problem_type": "Geometry",
"unit": null
} |
Let $T=\frac{1}{40}$. Given that $x, y$, and $z$ are real numbers such that $x+y=5, x^{2}-y^{2}=\frac{1}{T}$, and $x-z=-7$, compute $x+z$ | 20 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 3030,
"problem_type": "Algebra",
"unit": null
} |
A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagonals.
Suppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Find the maximum possible number of isosceles ... | 1003 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 1750,
"problem_type": "Combinatorics",
"unit": null
} |
Billy and Crystal each have a bag of 9 balls. The balls in each bag are numbered from 1 to 9. Billy and Crystal each remove one ball from their own bag. Let $b$ be the sum of the numbers on the balls remaining in Billy's bag. Let $c$ be the sum of the numbers on the balls remaining in Crystal's bag. Determine the proba... | \frac{7}{27} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2572,
"problem_type": "Combinatorics",
"unit": null
} |
Let $T=12$. Equilateral triangle $A B C$ is given with side length $T$. Points $D$ and $E$ are the midpoints of $\overline{A B}$ and $\overline{A C}$, respectively. Point $F$ lies in space such that $\triangle D E F$ is equilateral and $\triangle D E F$ lies in a plane perpendicular to the plane containing $\triangle A... | 108 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2910,
"problem_type": "Geometry",
"unit": null
} |
Five equilateral triangles are drawn in the plane so that no two sides of any of the triangles are parallel. Compute the maximum number of points of intersection among all five triangles. | 60 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2734,
"problem_type": "Combinatorics",
"unit": null
} |
Determine all possible values of $r$ such that the three term geometric sequence 4, $4 r, 4 r^{2}$ is also an arithmetic sequence.
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9, 11 is an arithmetic sequence.) | 1 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2266,
"problem_type": "Algebra",
"unit": null
} |
Given a regular 16-gon, extend three of its sides to form a triangle none of whose vertices lie on the 16-gon itself. Compute the number of noncongruent triangles that can be formed in this manner. | 11 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 3086,
"problem_type": "Geometry",
"unit": null
} |
Let $a, b, m, n$ be positive integers with $a m=b n=120$ and $a \neq b$. In the coordinate plane, let $A=(a, m), B=(b, n)$, and $O=(0,0)$. If $X$ is a point in the plane such that $A O B X$ is a parallelogram, compute the minimum area of $A O B X$. | 44 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2920,
"problem_type": "Geometry",
"unit": null
} |
The six sides of convex hexagon $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$ are colored red. Each of the diagonals of the hexagon is colored either red or blue. Compute the number of colorings such that every triangle $A_{i} A_{j} A_{k}$ has at least one red side. | 392 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2953,
"problem_type": "Geometry",
"unit": null
} |
Thurka bought some stuffed goats and some toy helicopters. She paid a total of $\$ 201$. She did not buy partial goats or partial helicopters. Each stuffed goat cost $\$ 19$ and each toy helicopter cost $\$ 17$. How many of each did she buy? | 7,4 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2369,
"problem_type": "Combinatorics",
"unit": null
} |
Determine all triples $(x, y, z)$ of real numbers that satisfy the following system of equations:
$$
\begin{aligned}
(x-1)(y-2) & =0 \\
(x-3)(z+2) & =0 \\
x+y z & =9
\end{aligned}
$$ | (1,-4,-2),(3,2,3),(13,2,-2) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2330,
"problem_type": "Algebra",
"unit": null
} |
Hexagon $A B C D E F$ has vertices $A(0,0), B(4,0), C(7,2), D(7,5), E(3,5)$, $F(0,3)$. What is the area of hexagon $A B C D E F$ ? | 29 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2504,
"problem_type": "Geometry",
"unit": null
} |
Compute the number of integers $n$ for which $2^{4}<8^{n}<16^{32}$. | 41 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 3044,
"problem_type": "Algebra",
"unit": null
} |
An ellipse in the first quadrant is tangent to both the $x$-axis and $y$-axis. One focus is at $(3,7)$, and the other focus is at $(d, 7)$. Compute $d$. | \frac{49}{3} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2890,
"problem_type": "Geometry",
"unit": null
} |
Suppose that $r$ and $s$ are the two roots of the equation $F_{k} x^{2}+F_{k+1} x+F_{k+2}=0$, where $F_{n}$ denotes the $n^{\text {th }}$ Fibonacci number. Compute the value of $(r+1)(s+1)$. | 2 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 3090,
"problem_type": "Algebra",
"unit": null
} |
Define $P(n)=n^{2}+n+1$. For any positive integers $a$ and $b$, the set
$$
\{P(a), P(a+1), P(a+2), \ldots, P(a+b)\}
$$
is said to be fragrant if none of its elements is relatively prime to the product of the other elements. Determine the smallest size of a fragrant set. | 6 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 1866,
"problem_type": "Number Theory",
"unit": null
} |
Let $T=\frac{1}{6084}$. Compute the least positive integer $n$ such that when a fair coin is flipped $n$ times, the probability of it landing heads on all $n$ flips is less than $T$. | 13 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2747,
"problem_type": "Combinatorics",
"unit": null
} |
Compute the smallest $n$ such that in the regular $n$-gon $A_{1} A_{2} A_{3} \cdots A_{n}, \mathrm{~m} \angle A_{1} A_{20} A_{13}<60^{\circ}$. | 37 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2676,
"problem_type": "Geometry",
"unit": null
} |
Let $N$ be a perfect square between 100 and 400 , inclusive. What is the only digit that cannot appear in $N$ ? | 7 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2960,
"problem_type": "Number Theory",
"unit": null
} |
Compute the base $b$ for which $253_{b} \cdot 341_{b}=\underline{7} \underline{4} \underline{X} \underline{Y} \underline{Z}_{b}$, for some base- $b$ digits $X, Y, Z$. | 20 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 3080,
"problem_type": "Number Theory",
"unit": null
} |
Determine all triples $(a, b, c)$ of positive integers for which $a b-c, b c-a$, and $c a-b$ are powers of 2 .
Explanation: A power of 2 is an integer of the form $2^{n}$, where $n$ denotes some nonnegative integer. | (2,2,2),(2,2,3),(2,3,2),(3,2,2),(2,6,11),(2,11,6),(6,2,11),(6,11,2),(11,2,6),(11,6,2),(3,5,7),(3,7,5),(5,3,7),(5,7,3),(7,3,5),(7,5,3) | {
"answer_type": "Tuple",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 1962,
"problem_type": "Number Theory",
"unit": null
} |
Eight people, including triplets Barry, Carrie and Mary, are going for a trip in four canoes. Each canoe seats two people. The eight people are to be randomly assigned to the four canoes in pairs. What is the probability that no two of Barry, Carrie and Mary will be in the same canoe? | \frac{4}{7} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2582,
"problem_type": "Combinatorics",
"unit": null
} |
Determine all integers $n \geqslant 1$ for which there exists a pair of positive integers $(a, b)$ such that no cube of a prime divides $a^{2}+b+3$ and
$$
\frac{a b+3 b+8}{a^{2}+b+3}=n
$$ | 2 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 1798,
"problem_type": "Number Theory",
"unit": null
} |
In a sequence of $n$ consecutive positive integers, where $n>1$, an element of the sequence is said to be cromulent if it is relatively prime to all other numbers in the sequence. Every element of a sequence with $n=2$ is cromulent because any two consecutive integers are relatively prime to each other.
Find the maximu... | 1,3 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2754,
"problem_type": "Algebra",
"unit": null
} |
In quadrilateral $A B C D, \mathrm{~m} \angle B+\mathrm{m} \angle D=270^{\circ}$. The circumcircle of $\triangle A B D$ intersects $\overline{C D}$ at point $E$, distinct from $D$. Given that $B C=4, C E=5$, and $D E=7$, compute the diameter of the circumcircle of $\triangle A B D$. | \sqrt{130} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2768,
"problem_type": "Geometry",
"unit": null
} |
Let $f(x)=2^{k x}+9$, where $k$ is a real number. If $f(3): f(6)=1: 3$, determine the value of $f(9)-f(3)$. | 210 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2242,
"problem_type": "Algebra",
"unit": null
} |
Oi-Lam tosses three fair coins and removes all of the coins that come up heads. George then tosses the coins that remain, if any. Determine the probability that George tosses exactly one head. | \frac{27}{64} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2377,
"problem_type": "Combinatorics",
"unit": null
} |
Compute the smallest positive integer base $b$ for which $16_{b}$ is prime and $97_{b}$ is a perfect square. | 53 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2671,
"problem_type": "Number Theory",
"unit": null
} |
Let $T=43$. Compute the positive integer $n \neq 17$ for which $\left(\begin{array}{c}T-3 \\ 17\end{array}\right)=\left(\begin{array}{c}T-3 \\ n\end{array}\right)$. | 23 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2777,
"problem_type": "Number Theory",
"unit": null
} |
Tanner has two identical dice. Each die has six faces which are numbered 2, 3, 5, $7,11,13$. When Tanner rolls the two dice, what is the probability that the sum of the numbers on the top faces is a prime number? | \frac{1}{6} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2454,
"problem_type": "Combinatorics",
"unit": null
} |
A two-digit number has the property that the square of its tens digit plus ten times its units digit equals the square of its units digit plus ten times its tens digit. Determine all two-digit numbers which have this property, and are prime numbers. | 11,19,37,73 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2443,
"problem_type": "Algebra",
"unit": null
} |
In the coordinate plane consider the set $S$ of all points with integer coordinates. For a positive integer $k$, two distinct points $A, B \in S$ will be called $k$-friends if there is a point $C \in S$ such that the area of the triangle $A B C$ is equal to $k$. A set $T \subset S$ will be called a $k$-clique if every ... | 180180 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2126,
"problem_type": "Combinatorics",
"unit": null
} |
Let $T=70$. Chef Selma is preparing a burrito menu. A burrito consists of: (1) a choice of chicken, beef, turkey, or no meat, (2) exactly one of three types of beans, (3) exactly one of two types of rice, and (4) exactly one of $K$ types of cheese. Compute the smallest value of $K$ such that Chef Selma can make at leas... | 3 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2661,
"problem_type": "Combinatorics",
"unit": null
} |
The three-digit positive integer $m$ is odd and has three distinct digits. If the hundreds digit of $m$ equals the product of the tens digit and ones (units) digit of $m$, what is $m$ ? | 623 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2321,
"problem_type": "Number Theory",
"unit": null
} |
Let $T=T N Y W R$. Compute $2^{\log _{T} 8}-8^{\log _{T} 2}$. | 0 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2720,
"problem_type": "Algebra",
"unit": null
} |
Let $T=2 \sqrt{2}$. In a right triangle, one leg has length $T^{2}$ and the other leg is 2 less than the hypotenuse. Compute the triangle's perimeter. | 40 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2650,
"problem_type": "Geometry",
"unit": null
} |
Let $n \geqslant 2$ be an integer. Consider an $n \times n$ chessboard divided into $n^{2}$ unit squares. We call a configuration of $n$ rooks on this board happy if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that for every happy configuration of rooks, we can find... | \lfloor\sqrt{n-1}\rfloor | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 1974,
"problem_type": "Combinatorics",
"unit": null
} |
Qing is twice as old as Rayna. Qing is 4 years younger than Paolo. The average age of Paolo, Qing and Rayna is 13. Determine their ages. | 7,14,18 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2277,
"problem_type": "Algebra",
"unit": null
} |
Rectangle $A R M L$ has length 125 and width 8. The rectangle is divided into 1000 squares of area 1 by drawing in gridlines parallel to the sides of $A R M L$. Diagonal $\overline{A M}$ passes through the interior of exactly $n$ of the 1000 unit squares. Compute $n$. | 132 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2976,
"problem_type": "Geometry",
"unit": null
} |
Let $n \geqslant 2$ be an integer, and let $f$ be a $4 n$-variable polynomial with real coefficients. Assume that, for any $2 n$ points $\left(x_{1}, y_{1}\right), \ldots,\left(x_{2 n}, y_{2 n}\right)$ in the plane, $f\left(x_{1}, y_{1}, \ldots, x_{2 n}, y_{2 n}\right)=0$ if and only if the points form the vertices of ... | 2n | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 1620,
"problem_type": "Algebra",
"unit": null
} |
Determine all possible values for the area of a right-angled triangle with one side length equal to 60 and with the property that its side lengths form an arithmetic sequence.
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, ... | 2400, 1350, 864 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2521,
"problem_type": "Number Theory",
"unit": null
} |
Let $S=\{1,2, \ldots, 20\}$, and let $f$ be a function from $S$ to $S$; that is, for all $s \in S, f(s) \in S$. Define the sequence $s_{1}, s_{2}, s_{3}, \ldots$ by setting $s_{n}=\sum_{k=1}^{20} \underbrace{(f \circ \cdots \circ f)}_{n}(k)$. That is, $s_{1}=f(1)+$ $\cdots+f(20), s_{2}=f(f(1))+\cdots+f(f(20)), s_{3}=f(... | 140 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2717,
"problem_type": "Algebra",
"unit": null
} |
Determine the largest real number $a$ such that for all $n \geqslant 1$ and for all real numbers $x_{0}, x_{1}, \ldots, x_{n}$ satisfying $0=x_{0}<x_{1}<x_{2}<\cdots<x_{n}$, we have
$$
\frac{1}{x_{1}-x_{0}}+\frac{1}{x_{2}-x_{1}}+\cdots+\frac{1}{x_{n}-x_{n-1}} \geqslant a\left(\frac{2}{x_{1}}+\frac{3}{x_{2}}+\cdots+\fr... | \frac{4}{9} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 1845,
"problem_type": "Algebra",
"unit": null
} |
Let $T=12$. Each interior angle of a regular $T$-gon has measure $d^{\circ}$. Compute $d$. | 150 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 3089,
"problem_type": "Geometry",
"unit": null
} |
A bag contains 20 lavender marbles, 12 emerald marbles, and some number of orange marbles. If the probability of drawing an orange marble in one try is $\frac{1}{y}$, compute the sum of all possible integer values of $y$. | 69 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 3038,
"problem_type": "Combinatorics",
"unit": null
} |
A regular hexagon has side length 1. Compute the average of the areas of the 20 triangles whose vertices are vertices of the hexagon. | \frac{9 \sqrt{3}}{20} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2706,
"problem_type": "Geometry",
"unit": null
} |
The equation $2^{x+2} 5^{6-x}=10^{x^{2}}$ has two real solutions. Determine these two solutions. | 2,-\log _{10} 250 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2574,
"problem_type": "Algebra",
"unit": null
} |
Define $\log ^{*}(n)$ to be the smallest number of times the log function must be iteratively applied to $n$ to get a result less than or equal to 1 . For example, $\log ^{*}(1000)=2$ since $\log 1000=3$ and $\log (\log 1000)=\log 3=0.477 \ldots \leq 1$. Let $a$ be the smallest integer such that $\log ^{*}(a)=3$. Compu... | 9 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2951,
"problem_type": "Algebra",
"unit": null
} |
Given an arbitrary finite sequence of letters (represented as a word), a subsequence is a sequence of one or more letters that appear in the same order as in the original sequence. For example, $N, C T, O T T$, and CONTEST are subsequences of the word CONTEST, but NOT, ONSET, and TESS are not. Assuming the standard Eng... | 3851 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2955,
"problem_type": "Combinatorics",
"unit": null
} |
In an arithmetic sequence with 5 terms, the sum of the squares of the first 3 terms equals the sum of the squares of the last 2 terms. If the first term is 5 , determine all possible values of the fifth term.
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by... | -5,7 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2421,
"problem_type": "Algebra",
"unit": null
} |
A circle $\omega$ of radius 1 is given. A collection $T$ of triangles is called good, if the following conditions hold:
(i) each triangle from $T$ is inscribed in $\omega$;
(ii) no two triangles from $T$ have a common interior point.
Determine all positive real numbers $t$ such that, for each positive integer $n$, t... | t(0,4] | {
"answer_type": "Interval",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2045,
"problem_type": "Geometry",
"unit": null
} |
If $f(x)=2 x+1$ and $g(f(x))=4 x^{2}+1$, determine an expression for $g(x)$. | g(x)=x^2-2x+2 | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2371,
"problem_type": "Algebra",
"unit": null
} |
Determine the smallest positive real number $k$ with the following property.
Let $A B C D$ be a convex quadrilateral, and let points $A_{1}, B_{1}, C_{1}$ and $D_{1}$ lie on sides $A B, B C$, $C D$ and $D A$, respectively. Consider the areas of triangles $A A_{1} D_{1}, B B_{1} A_{1}, C C_{1} B_{1}$, and $D D_{1} C_{1... | k=1 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 1675,
"problem_type": "Geometry",
"unit": null
} |
A rectangular box has dimensions $8 \times 10 \times 12$. Compute the fraction of the box's volume that is not within 1 unit of any of the box's faces. | \frac{1}{2} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2855,
"problem_type": "Geometry",
"unit": null
} |
Let $A, B$, and $C$ be randomly chosen (not necessarily distinct) integers between 0 and 4 inclusive. Pat and Chris compute the value of $A+B \cdot C$ by two different methods. Pat follows the proper order of operations, computing $A+(B \cdot C)$. Chris ignores order of operations, choosing instead to compute $(A+B) \c... | \frac{9}{25} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2599,
"problem_type": "Combinatorics",
"unit": null
} |
Suppose that $n$ is a positive integer and that the value of $\frac{n^{2}+n+15}{n}$ is an integer. Determine all possible values of $n$. | 1, 3, 5, 15 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2323,
"problem_type": "Number Theory",
"unit": null
} |
In triangle $A B C, A B=4, B C=6$, and $A C=8$. Squares $A B Q R$ and $B C S T$ are drawn external to and lie in the same plane as $\triangle A B C$. Compute $Q T$. | 2 \sqrt{10} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2888,
"problem_type": "Geometry",
"unit": null
} |
If $\frac{(x-2013)(y-2014)}{(x-2013)^{2}+(y-2014)^{2}}=-\frac{1}{2}$, what is the value of $x+y$ ? | 4027 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2345,
"problem_type": "Algebra",
"unit": null
} |
In the plane, 2013 red points and 2014 blue points are marked so that no three of the marked points are collinear. One needs to draw $k$ lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors.
Find the mi... | 2013 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2064,
"problem_type": "Combinatorics",
"unit": null
} |
If $T=x^{2}+\frac{1}{x^{2}}$, determine the values of $b$ and $c$ so that $x^{6}+\frac{1}{x^{6}}=T^{3}+b T+c$ for all non-zero real numbers $x$. | -3,0 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": true,
"problem_idx": 2317,
"problem_type": "Algebra",
"unit": null
} |
Let $n$ be a positive integer. Harry has $n$ coins lined up on his desk, each showing heads or tails. He repeatedly does the following operation: if there are $k$ coins showing heads and $k>0$, then he flips the $k^{\text {th }}$ coin over; otherwise he stops the process. (For example, the process starting with THT wou... | \frac{1}{4} n(n+1) | {
"answer_type": "Expression",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 1914,
"problem_type": "Combinatorics",
"unit": null
} |
Let $T=1$. In parallelogram $A B C D, \frac{A B}{B C}=T$. Given that $M$ is the midpoint of $\overline{A B}$ and $P$ and $Q$ are the trisection points of $\overline{C D}$, compute $\frac{[A B C D]}{[M P Q]}$. | 6 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 3021,
"problem_type": "Geometry",
"unit": null
} |
Compute the value of
$$
\sin \left(6^{\circ}\right) \cdot \sin \left(12^{\circ}\right) \cdot \sin \left(24^{\circ}\right) \cdot \sin \left(42^{\circ}\right)+\sin \left(12^{\circ}\right) \cdot \sin \left(24^{\circ}\right) \cdot \sin \left(42^{\circ}\right) \text {. }
$$ | \frac{1}{16} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2984,
"problem_type": "Algebra",
"unit": null
} |
Let $T=1$. Dennis and Edward each take 48 minutes to mow a lawn, and Shawn takes 24 minutes to mow a lawn. Working together, how many lawns can Dennis, Edward, and Shawn mow in $2 \cdot T$ hours? (For the purposes of this problem, you may assume that after they complete mowing a lawn, they immediately start mowing the ... | 10 | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 3099,
"problem_type": "Algebra",
"unit": null
} |
Three different numbers are chosen at random from the set $\{1,2,3,4,5\}$.
The numbers are arranged in increasing order.
What is the probability that the resulting sequence is an arithmetic sequence?
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding... | \frac{2}{5} | {
"answer_type": "Numerical",
"dataset": "olympiadbench",
"difficulty": "olympiad",
"is_multiple_answer": false,
"problem_idx": 2249,
"problem_type": "Combinatorics",
"unit": null
} |
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