problem stringlengths 10 5.15k | answer dict |
|---|---|
Three friends are in the park. Bob and Clarise are standing at the same spot and Abe is standing 10 m away. Bob chooses a random direction and walks in this direction until he is 10 m from Clarise. What is the probability that Bob is closer to Abe than Clarise is to Abe? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $n$ students in the math club. When grouped in 4s, there is one incomplete group. When grouped in 3s, there are 3 more complete groups than with 4s, and one incomplete group. When grouped in 2s, there are 5 more complete groups than with 3s, and one incomplete group. What is the sum of the digits of $n^{2}-n$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many words are there in a language that are 10 letters long and begin with a vowel, given that the language uses only the letters A, B, C, D, and E, where A and E are vowels, and B, C, and D are consonants, and a word does not include the same letter twice in a row or two vowels in a row? | {
"answer": "199776",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of $x$ if $P Q S$ is a straight line and $\angle P Q R=110^{\circ}$? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the 200 students surveyed said that their favourite food was sandwiches, given the circle graph results? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, $XY=XZ$ and $W$ is on $XZ$ such that $XW=WY=YZ$. What is the measure of $\angle XYW$? | {
"answer": "36^{\\circ}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A tetrahedron of spheres is formed with thirteen layers and each sphere has a number written on it. The top sphere has a 1 written on it and each of the other spheres has written on it the number equal to the sum of the numbers on the spheres in the layer above with which it is in contact. What is the sum of the numbers on all of the internal spheres? | {
"answer": "772626",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular piece of paper $P Q R S$ has $P Q=20$ and $Q R=15$. The piece of paper is glued flat on the surface of a large cube so that $Q$ and $S$ are at vertices of the cube. What is the shortest distance from $P$ to $R$, as measured through the cube? | {
"answer": "18.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Storage space on a computer is measured in gigabytes (GB) and megabytes (MB), where $1 \mathrm{~GB} = 1024 \mathrm{MB}$. Julia has an empty 300 GB hard drive and puts 300000 MB of data onto it. How much storage space on the hard drive remains empty? | {
"answer": "7200 \\mathrm{MB}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The regular price for a bicycle is $\$320$. The bicycle is on sale for $20\%$ off. The regular price for a helmet is $\$80$. The helmet is on sale for $10\%$ off. If Sandra bought both items on sale, what is her percentage savings on the total purchase? | {
"answer": "18\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $N = 3x + 4y + 5z$, where $x$ equals 1 or -1, and $y$ equals 1 or -1, and $z$ equals 1 or -1. How many of the following statements are true? - $N$ can equal 0. - $N$ is always odd. - $N$ cannot equal 4. - $N$ is always even. | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many such nine-digit positive integers can Ricardo make if he wants to arrange three 1s, three 2s, two 3s, and one 4 with the properties that there is at least one 1 before the first 2, at least one 2 before the first 3, and at least one 3 before the 4, and no digit 2 can be next to another 2? | {
"answer": "254",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two numbers $a$ and $b$ with $0 \leq a \leq 1$ and $0 \leq b \leq 1$ are chosen at random. The number $c$ is defined by $c=2a+2b$. The numbers $a, b$ and $c$ are each rounded to the nearest integer to give $A, B$ and $C$, respectively. What is the probability that $2A+2B=C$? | {
"answer": "\\frac{7}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of all numbers $q$ which can be written in the form $q=\frac{a}{b}$ where $a$ and $b$ are positive integers with $b \leq 10$ and for which there are exactly 19 integers $n$ that satisfy $\sqrt{q}<n<q$? | {
"answer": "777.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate \[ \lim_{x \to 1^-} \prod_{n=0}^\infty \left(\frac{1 + x^{n+1}}{1 + x^n}\right)^{x^n}. \] | {
"answer": "\\frac{2}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For every real number $x$, what is the value of the expression $(x+1)^{2} - x^{2}$? | {
"answer": "2x + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $PQRS TUVW$ is a regular octagon. There are 70 ways in which four of its sides can be chosen at random. If four of its sides are chosen at random and each of these sides is extended infinitely in both directions, what is the probability that they will meet to form a quadrilateral that contains the octagon? | {
"answer": "\\frac{19}{35}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The set $S$ consists of 9 distinct positive integers. The average of the two smallest integers in $S$ is 5. The average of the two largest integers in $S$ is 22. What is the greatest possible average of all of the integers of $S$? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle $W X Y Z$ has $W X=4, W Z=3$, and $Z V=3$. The rectangle is curled without overlapping into a cylinder so that sides $W Z$ and $X Y$ touch each other. In other words, $W$ touches $X$ and $Z$ touches $Y$. The shortest distance from $W$ to $V$ through the inside of the cylinder can be written in the form $\sqrt{\frac{a+b \pi^{2}}{c \pi^{2}}}$ where $a, b$ and $c$ are positive integers. What is the smallest possible value of $a+b+c$? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cube $A B C D E F G H$ has edge length 100. Point $P$ is on $A B$, point $Q$ is on $A D$, and point $R$ is on $A F$, as shown, so that $A P=x, A Q=x+1$ and $A R=\frac{x+1}{2 x}$ for some integer $x$. For how many integers $x$ is the volume of triangular-based pyramid $A P Q R$ between $0.04 \%$ and $0.08 \%$ of the volume of cube $A B C D E F G H$? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x = 2y$ and $y \neq 0$, what is the value of $(x-y)(2x+y)$? | {
"answer": "5y^{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alain and Louise are driving on a circular track with radius 25 km. Alain leaves the starting line first, going clockwise at 80 km/h. Fifteen minutes later, Louise leaves the same starting line, going counterclockwise at 100 km/h. For how many hours will Louise have been driving when they pass each other for the fourth time? | {
"answer": "\\\\frac{10\\\\pi-1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
After a fair die with faces numbered 1 to 6 is rolled, the number on the top face is $x$. What is the most likely outcome? | {
"answer": "x > 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of $| \sin x + \cos x + \tan x + \cot x + \sec x + \csc x |$ for real numbers $x$. | {
"answer": "2\\sqrt{2} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S = \{1, 2, \dots, n\}$ for some integer $n > 1$. Say a permutation $\pi$ of $S$ has a \emph{local maximum} at $k \in S$ if \begin{enumerate} \item[(i)] $\pi(k) > \pi(k+1)$ for $k=1$; \item[(ii)] $\pi(k-1) < \pi(k)$ and $\pi(k) > \pi(k+1)$ for $1 < k < n$; \item[(iii)] $\pi(k-1) < \pi(k)$ for $k=n$. \end{enumerate} (For example, if $n=5$ and $\pi$ takes values at $1, 2, 3, 4, 5$ of $2, 1, 4, 5, 3$, then $\pi$ has a local maximum of 2 at $k=1$, and a local maximum of 5 at $k=4$.) What is the average number of local maxima of a permutation of $S$, averaging over all permutations of $S$? | {
"answer": "\\frac{n+1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \sqrt{2k})$. Evaluate
\[
\sum_{k=1}^\infty (-1)^{k-1} \frac{A(k)}{k}.
\] | {
"answer": "\\frac{\\pi^2}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Snacks are purchased for 17 soccer players. Juice boxes come in packs of 3 and cost $2.00 per pack. Apples come in bags of 5 and cost $4.00 per bag. What is the minimum amount of money that Danny spends to ensure every player gets a juice box and an apple? | {
"answer": "\\$28.00",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a \rfloor, \lfloor 2a \rfloor) = 0$ for all real numbers $a$. (Note: $\lfloor \nu \rfloor$ is the greatest integer less than or equal to $\nu$.) | {
"answer": "(y-2x)(y-2x-1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For all $n \geq 1$, let \[ a_n = \sum_{k=1}^{n-1} \frac{\sin \left( \frac{(2k-1)\pi}{2n} \right)}{\cos^2 \left( \frac{(k-1)\pi}{2n} \right) \cos^2 \left( \frac{k\pi}{2n} \right)}. \] Determine \[ \lim_{n \to \infty} \frac{a_n}{n^3}. \] | {
"answer": "\\frac{8}{\\pi^3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many odd integers $k$ between 0 and 100 does the equation $2^{4m^{2}}+2^{m^{2}-n^{2}+4}=2^{k+4}+2^{3m^{2}+n^{2}+k}$ have exactly two pairs of positive integers $(m, n)$ that are solutions? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of $a, b$ and $c$ is equal to a number from the list $3^{1}, 3^{2}, 3^{3}, 3^{4}, 3^{5}, 3^{6}, 3^{7}, 3^{8}$. There are $N$ triples $(a, b, c)$ with $a \leq b \leq c$ for which each of $\frac{ab}{c}, \frac{ac}{b}$ and $\frac{bc}{a}$ is equal to an integer. What is the value of $N$? | {
"answer": "86",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that \[ \binom{m}{n-1} > \binom{m-1}{n}. \] Evaluate \[ \lim_{n \to \infty} \frac{M(n)}{n}. \] | {
"answer": "\\frac{3+\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pizza is cut into 10 pieces. Two of the pieces are each \(\frac{1}{24}\) of the whole pizza, four are each \(\frac{1}{12}\), two are each \(\frac{1}{8}\), and two are each \(\frac{1}{6}\). A group of \(n\) friends share the pizza by distributing all of these pieces. They do not cut any of these pieces. Each of the \(n\) friends receives, in total, an equal fraction of the whole pizza. What is the sum of the values of \(n\) with \(2 \leq n \leq 10\) for which this is not possible? | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral. | {
"answer": "1 - \\frac{35}{12 \\pi^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,\dots,30$. How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? | {
"answer": "61917364224",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the number line, points $M$ and $N$ divide $L P$ into three equal parts. What is the value at $M$? | {
"answer": "\\frac{1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $P$ is on the $y$-axis with $y$-coordinate greater than 0 and less than 100. A circle is drawn through $P, Q(4,4)$ and $O(0,0)$. How many possible positions for $P$ are there so that the radius of this circle is an integer? | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all positive integers $n, k_1, \dots, k_n$ such that $k_1 + \cdots + k_n = 5n-4$ and \[ \frac{1}{k_1} + \cdots + \frac{1}{k_n} = 1. \] | {
"answer": "n = 1, k_1 = 1; n = 3, (k_1,k_2,k_3) = (2,3,6); n = 4, (k_1,k_2,k_3,k_4) = (4,4,4,4)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many values of $n$ with $3 \leq n \leq 12$ can a Fano table be created? | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area of the region \( \mathcal{R} \) formed by all points \( L(r, t) \) with \( 0 \leq r \leq 10 \) and \( 0 \leq t \leq 10 \) such that the area of triangle \( JKL \) is less than or equal to 10, where \( J(2,7) \) and \( K(5,3) \)? | {
"answer": "\\frac{313}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by \[ q(x) = \sum_{k=1}^{p-1} a_k x^k, \] where \[ a_k = k^{(p-1)/2} \mod{p}. \] Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$. | {
"answer": "\\frac{p-1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least number of gumballs that Wally must buy to guarantee that he receives 3 gumballs of the same colour? | {
"answer": "8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a rhombus $P Q R S$ with $P Q=Q R=R S=S P=S Q=6$ and $P T=R T=14$, what is the length of $S T$? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each real number $x$, let
\[
f(x) = \sum_{n\in S_x} \frac{1}{2^n},
\]
where $S_x$ is the set of positive integers $n$ for which $\lfloor nx \rfloor$ is even. What is the largest real number $L$ such that $f(x) \geq L$ for all $x \in [0,1)$? (As usual, $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.) | {
"answer": "4/7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence $y_1,y_2,\dots,y_k$ of real numbers is called \emph{zigzag} if $k=1$, or if $y_2-y_1, y_3-y_2, \dots, y_k-y_{k-1}$ are nonzero and alternate in sign. Let $X_1,X_2,\dots,X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a(X_1,X_2,\dots,X_n)$ be the largest value of $k$ for which there exists an increasing sequence of integers $i_1,i_2,\dots,i_k$ such that $X_{i_1},X_{i_2},\dots,X_{i_k}$ is zigzag. Find the expected value of $a(X_1,X_2,\dots,X_n)$ for $n \geq 2$. | {
"answer": "\\frac{2n+2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule. Alice's goal is to maximize the number of uncovered squares when the game ends; Bob's goal is to minimize it. What is the greatest number of uncovered squares that Alice can ensure at the end of the game, no matter how Bob plays? | {
"answer": "290",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Dolly, Molly, and Polly each can walk at $6 \mathrm{~km} / \mathrm{h}$. Their one motorcycle, which travels at $90 \mathrm{~km} / \mathrm{h}$, can accommodate at most two of them at once. What is true about the smallest possible time $t$ for all three of them to reach a point 135 km away? | {
"answer": "t < 3.9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider an $m$-by-$n$ grid of unit squares, indexed by $(i,j)$ with $1 \leq i \leq m$ and $1 \leq j \leq n$. There are $(m-1)(n-1)$ coins, which are initially placed in the squares $(i,j)$ with $1 \leq i \leq m-1$ and $1 \leq j \leq n-1$. If a coin occupies the square $(i,j)$ with $i \leq m-1$ and $j \leq n-1$ and the squares $(i+1,j), (i,j+1)$, and $(i+1,j+1)$ are unoccupied, then a legal move is to slide the coin from $(i,j)$ to $(i+1,j+1)$. How many distinct configurations of coins can be reached starting from the initial configuration by a (possibly empty) sequence of legal moves? | {
"answer": "\\binom{m+n-2}{m-1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \geq 0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2 + y^2 = 2^k$ for some integer $k \leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square. | {
"answer": "5n+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangle $\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle. Let $\alpha$ and $\beta$ be the angles at the vertices $A$ and $B$, respectively. Suppose that the segment $IG$ is parallel to $AB$ and that $\beta = 2 \tan^{-1} (1/3)$. Find $\alpha$. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $N$ is a positive integer between 1000000 and 10000000, inclusive, what is the maximum possible value for the sum of the digits of $25 \times N$? | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any positive integer $n$, let \langle n\rangle denote the closest integer to \sqrt{n}. Evaluate
\[\sum_{n=1}^\infty \frac{2^{\langle n\rangle}+2^{-\langle n\rangle}}{2^n}.\] | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $t_{n}$ equal the integer closest to $\sqrt{n}$. What is the sum $\frac{1}{t_{1}}+\frac{1}{t_{2}}+\frac{1}{t_{3}}+\frac{1}{t_{4}}+\cdots+\frac{1}{t_{2008}}+\frac{1}{t_{2009}}+\frac{1}{t_{2010}}$? | {
"answer": "88 \\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be the roots in the complex plane of the polynomial \[ P(z) = \sum_{k=0}^{2019} b_k z^k. \] Let $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\dots,z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu \geq M$ for all choices of $b_0,b_1,\dots, b_{2019}$ that satisfy \[ 1 \leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019. \] | {
"answer": "2019^{-1/2019}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
\[
\log_2 \left( \prod_{a=1}^{2015} \prod_{b=1}^{2015} (1+e^{2\pi i a b/2015}) \right)
\]
Here $i$ is the imaginary unit (that is, $i^2=-1$). | {
"answer": "13725",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate \[ \sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \sum_{n=0}^\infty \frac{1}{k2^n + 1}. \] | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A \cup B = S$, $A \cap B = \emptyset$, and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_S$ be the number of linear partitions of $S$. For each positive integer $n$, find the maximum of $L_S$ over all sets $S$ of $n$ points. | {
"answer": "\\binom{n}{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the maximum value of the sum
\[S = \sum_{n=1}^\infty \frac{n}{2^n} (a_1 a_2 \cdots a_n)^{1/n}\]
over all sequences $a_1, a_2, a_3, \cdots$ of nonnegative real numbers satisfying
\[\sum_{k=1}^\infty a_k = 1.\] | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1? | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest constant $C$ such that for every real polynomial $P(x)$ of degree 3 that has a root in the interval $[0,1]$, \[ \int_0^1 \left| P(x) \right|\,dx \leq C \max_{x \in [0,1]} \left| P(x) \right|. \] | {
"answer": "\\frac{5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[ X \right] = 1$, $E\left[ X^2 \right] = 2$, and $E \left[ X^3 \right] = 5$. Determine the smallest possible value of the probability of the event $X=0$. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be a positive integer. What is the largest $k$ for which there exist $n \times n$ matrices $M_1, \dots, M_k$ and $N_1, \dots, N_k$ with real entries such that for all $i$ and $j$, the matrix product $M_i N_j$ has a zero entry somewhere on its diagonal if and only if $i \neq j$? | {
"answer": "n^n",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle, with the vertices around the circumference in the given order. Given that the polygon $P_1P_3P_5P_7$ is a square of area 5, and the polygon $P_2P_4P_6P_8$ is a rectangle of area 4, find the maximum possible area of the octagon. | {
"answer": "3\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be a positive integer. Determine, in terms of $n$, the largest integer $m$ with the following property: There exist real numbers $x_1,\dots,x_{2n}$ with $-1 < x_1 < x_2 < \cdots < x_{2n} < 1$ such that the sum of the lengths of the $n$ intervals \[ [x_1^{2k-1}, x_2^{2k-1}], [x_3^{2k-1},x_4^{2k-1}], \dots, [x_{2n-1}^{2k-1}, x_{2n}^{2k-1}] \] is equal to 1 for all integers $k$ with $1 \leq k \leq m$. | {
"answer": "n",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1) = 1$, $c(2n) = c(n)$, and $c(2n+1) = (-1)^n c(n)$. Find the value of \[ \sum_{n=1}^{2013} c(n) c(n+2). \] | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Say that a polynomial with real coefficients in two variables, $x,y$, is \emph{balanced} if
the average value of the polynomial on each circle centered at the origin is $0$.
The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$.
Find the dimension of $V$. | {
"answer": "2020050",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n$; that is to say, $p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0$ for some real coefficients $a_0, \dots, a_{2n-1}$. Suppose that $p(1/k) = k^2$ for all integers $k$ such that $1 \leq |k| \leq n$. Find all other real numbers $x$ for which $p(1/x) = x^2$. | {
"answer": "\\pm 1/n!",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly 50 of her first 100 shots? | {
"answer": "\\(\\frac{1}{99}\\)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a_0 = 5/2$ and $a_k = a_{k-1}^2 - 2$ for $k \geq 1$. Compute \[ \prod_{k=0}^\infty \left(1 - \frac{1}{a_k} \right) \] in closed form. | {
"answer": "\\frac{3}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $k$ be a positive integer. Suppose that the integers $1, 2, 3, \dots, 3k+1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by 3? Your answer should be in closed form, but may include factorials. | {
"answer": "\\frac{k!(k+1)!}{(3k+1)(2k)!}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers \[ N = a + (a+1) +(a+2) + \cdots + (a+k-1) \] for $k=2017$ but for no other values of $k>1$. Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of these expressions? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A$ be a $2n \times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be 0 or 1, each with probability $1/2$. Find the expected value of $\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A$. | {
"answer": "\\frac{(2n)!}{4^n n!}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties: \begin{enumerate} \item[(a)] $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1,\dots,t_n$; \item[(b)] $f(t_0) = 1/2$; \item[(c)] $\lim_{t \to t_k^+} f'(t) = 0$ for $0 \leq k \leq n$; \item[(d)] For $0 \leq k \leq n-1$, we have $f''(t) = k+1$ when $t_k < t< t_{k+1}$, and $f''(t) = n+1$ when $t>t_n$. \end{enumerate} Considering all choices of $n$ and $t_0,t_1,\dots,t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f(t_0+T) = 2023$? | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard? | {
"answer": "2 - \\frac{6}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the subtraction shown, $K, L, M$, and $N$ are digits. What is the value of $K+L+M+N$? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all pairs of real numbers $(x,y)$ satisfying the system of equations
\begin{align*}
\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2) \\
\frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4).
\end{align*} | {
"answer": "x = (3^{1/5}+1)/2, y = (3^{1/5}-1)/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $Z$ denote the set of points in $\mathbb{R}^n$ whose coordinates are 0 or 1. (Thus $Z$ has $2^n$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^n$.) Given a vector subspace $V$ of $\mathbb{R}^n$, let $Z(V)$ denote the number of members of $Z$ that lie in $V$. Let $k$ be given, $0 \leq k \leq n$. Find the maximum, over all vector subspaces $V \subseteq \mathbb{R}^n$ of dimension $k$, of the number of points in $V \cap Z$. | {
"answer": "2^k",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be given, $n \geq 4$, and suppose that $P_1, P_2, \dots, P_n$ are $n$ randomly, independently and uniformly, chosen points on a circle. Consider the convex $n$-gon whose vertices are the $P_i$. What is the probability that at least one of the vertex angles of this polygon is acute? | {
"answer": "n(n-2) 2^{-n+1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2,\dots,n\}$, let $s(i,j)$ be the number of pairs $(a,b)$ of nonnegative integers satisfying $ai +bj=n$. Let $S$ be the $n$-by-$n$ matrix whose $(i,j)$ entry is $s(i,j)$. For example, when $n=5$, we have $S = \begin{bmatrix} 6 & 3 & 2 & 2 & 2 \\ 3 & 0 & 1 & 0 & 1 \\ 2 & 1 & 0 & 0 & 1 \\ 2 & 0 & 0 & 0 & 1 \\ 2 & 1 & 1 & 1 & 2 \end{bmatrix}$. Compute the determinant of $S$. | {
"answer": "(-1)^{\\lceil n/2 \\rceil-1} 2 \\lceil \\frac{n}{2} \\rceil",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $X_1, X_2, \dots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S = \sum_{i=1}^k X_i/2^i$, where $k$ is the least positive integer such that $X_k < X_{k+1}$, or $k = \infty$ if there is no such integer. Find the expected value of $S$. | {
"answer": "2e^{1/2}-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $2.4 \times 10^{8}$ is doubled, what is the result? | {
"answer": "4.8 \\times 10^{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\pi\colon \{1,2,\dots,n\} \to \{1,2,\dots,n\}$ such that $\pi(\pi(k)) \equiv mk \pmod{n}$ for all $k \in \{1,2,\dots,n\}$. | {
"answer": "n = 1 \\text{ or } n \\equiv 2 \\pmod{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of ordered $64$-tuples $(x_0,x_1,\dots,x_{63})$ such that $x_0,x_1,\dots,x_{63}$ are distinct elements of $\{1,2,\dots,2017\}$ and \[ x_0 + x_1 + 2x_2 + 3x_3 + \cdots + 63 x_{63} \] is divisible by 2017. | {
"answer": "$\\frac{2016!}{1953!}- 63! \\cdot 2016$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $F_m$ be the $m$th Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_m = F_{m-1} + F_{m-2}$ for all $m \geq 3$. Let $p(x)$ be the polynomial of degree $1008$ such that $p(2n+1) = F_{2n+1}$ for $n=0,1,2,\dots,1008$. Find integers $j$ and $k$ such that $p(2019) = F_j - F_k$. | {
"answer": "(j,k) = (2019, 1010)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ has an area 1. Points $E,F,G$ lie, respectively, on sides $BC$, $CA$, $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$. | {
"answer": "\\frac{7 - 3 \\sqrt{5}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest positive real number $r$ such that there exist differentiable functions $f\colon \mathbb{R} \to \mathbb{R}$ and $g\colon \mathbb{R} \to \mathbb{R}$ satisfying \begin{enumerate} \item[(a)] $f(0) > 0$, \item[(b)] $g(0) = 0$, \item[(c)] $|f'(x)| \leq |g(x)|$ for all $x$, \item[(d)] $|g'(x)| \leq |f(x)|$ for all $x$, and \item[(e)] $f(r) = 0$. \end{enumerate} | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $k$ be an integer greater than 1. Suppose $a_0 > 0$, and define \[a_{n+1} = a_n + \frac{1}{\sqrt[k]{a_n}}\] for $n > 0$. Evaluate \[\lim_{n \to \infty} \frac{a_n^{k+1}}{n^k}.\] | {
"answer": "\\left( \\frac{k+1}{k} \\right)^k",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a school fundraising campaign, $25\%$ of the money donated came from parents. The rest of the money was donated by teachers and students. The ratio of the amount of money donated by teachers to the amount donated by students was $2:3$. What is the ratio of the amount of money donated by parents to the amount donated by students? | {
"answer": "5:9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For every positive real number $x$, let
\[g(x) = \lim_{r \to 0} ((x+1)^{r+1} - x^{r+1})^{\frac{1}{r}}.\]
Find $\lim_{x \to \infty} \frac{g(x)}{x}$. | {
"answer": "e",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the perimeter of the shaded region in a \( 3 \times 3 \) grid where some \( 1 \times 1 \) squares are shaded? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 \cdot n$. What is the minimum value of $k(n)$? | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square has side length 5. In how many different locations can point $X$ be placed so that the distances from $X$ to the four sides of the square are $1,2,3$, and 4? | {
"answer": "8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ such that \[ f(x) + f\left( 1 - \frac{1}{x} \right) = \arctan x \] for all real $x \neq 0$. (As usual, $y = \arctan x$ means $-\pi/2 < y < \pi/2$ and $\tan y = x$.) Find \[ \int_0^1 f(x)\,dx. \] | {
"answer": "\\frac{3\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be an integer with $n \geq 2$. Over all real polynomials $p(x)$ of degree $n$, what is the largest possible number of negative coefficients of $p(x)^2$? | {
"answer": "2n-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A_1B_1C_1D_1$ be an arbitrary convex quadrilateral. $P$ is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point $P$ is acute. We recursively define points $A_k,B_k,C_k,D_k$ symmetric to $P$ with respect to lines $A_{k-1}B_{k-1}, B_{k-1}C_{k-1}, C_{k-1}D_{k-1},D_{k-1}A_{k-1}$ respectively for $k\ge 2$.
Consider the sequence of quadrilaterals $A_iB_iC_iD_i$.
i) Among the first 12 quadrilaterals, which are similar to the 1997th quadrilateral and which are not?
ii) Suppose the 1997th quadrilateral is cyclic. Among the first 12 quadrilaterals, which are cyclic and which are not? | {
"answer": "1, 5, 9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ A$ and $ B$ play the following game with a polynomial of degree at least 4:
\[ x^{2n} \plus{} \_x^{2n \minus{} 1} \plus{} \_x^{2n \minus{} 2} \plus{} \ldots \plus{} \_x \plus{} 1 \equal{} 0
\]
$ A$ and $ B$ take turns to fill in one of the blanks with a real number until all the blanks are filled up. If the resulting polynomial has no real roots, $ A$ wins. Otherwise, $ B$ wins. If $ A$ begins, which player has a winning strategy? | {
"answer": "B",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equation $a^3 + b^3 + c^3 = 2001$ in positive integers. | {
"answer": "\\[\n\\boxed{(10,10,1), (10,1,10), (1,10,10)}\n\\]",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
( Ricky Liu ) For what values of $k > 0$ is it possible to dissect a $1 \times k$ rectangle into two similar, but incongruent, polygons? | {
"answer": "\\[ k \\neq 1 \\]",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fat coin is one which, when tossed, has a $2 / 5$ probability of being heads, $2 / 5$ of being tails, and $1 / 5$ of landing on its edge. Mr. Fat starts at 0 on the real line. Every minute, he tosses a fat coin. If it's heads, he moves left, decreasing his coordinate by 1; if it's tails, he moves right, increasing his coordinate by 1. If the coin lands on its edge, he moves back to 0. If Mr. Fat does this ad infinitum, what fraction of his time will he spend at 0? | {
"answer": "\\[\n\\frac{1}{3}\n\\]",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a $4 \times 4$ grid of squares. Aziraphale and Crowley play a game on this grid, alternating turns, with Aziraphale going first. On Aziraphales turn, he may color any uncolored square red, and on Crowleys turn, he may color any uncolored square blue. The game ends when all the squares are colored, and Aziraphales score is the area of the largest closed region that is entirely red. If Aziraphale wishes to maximize his score, Crowley wishes to minimize it, and both players play optimally, what will Aziraphales score be? | {
"answer": "\\[ 6 \\]",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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