Dataset Viewer
problem
stringlengths 10
7.44k
| answer
stringlengths 1
270
| difficulty
stringclasses 8
values |
|---|---|---|
The front tires of a car wear out after 25,000 km, and the rear tires wear out after 15,000 km. When should the tires be swapped so that they wear out at the same time? | 9375 | 7/8 |
The edges of tetrahedron $ABCD$ are given as $AB=c$, $BC=a$, $CA=b$, $DA=a_{1}$, $DB=b_{1}$, and $DC=c_{1}$. Let $h$ be the length of the median line from vertex $D$. Prove that
$$
h^{2}=\frac{1}{3}\left(a_{1}^{2}+b_{1}^{2}+c_{1}^{2}\right)-\frac{1}{9}\left(a^{2}+b^{2}+c^{2}\right)
$$ | ^2=\frac{1}{3}(a_1^2+b_1^2+c_1^2)-\frac{1}{9}(^2+b^2+^2) | 5/8 |
The largest three-digit number divided by an integer, with the quotient rounded to one decimal place being 2.5, will have the smallest divisor as: | 392 | 6/8 |
Mr. Lee V. Soon starts his morning commute at 7:00 AM to arrive at work by 8:00 AM. If he drives at an average speed of 30 miles per hour, he is late by 5 minutes, and if he drives at an average speed of 70 miles per hour, he is early by 4 minutes. Find the speed he needs to maintain to arrive exactly at 8:00 AM. | 32.5 | 1/8 |
Suppose convex hexagon $ \text{HEXAGN}$ has $ 120^\circ$ -rotational symmetry about a point $ P$ —that is, if you rotate it $ 120^\circ$ about $ P$ , it doesn't change. If $ PX\equal{}1$ , find the area of triangle $ \triangle{GHX}$ . | \frac{\sqrt{3}}{4} | 1/8 |
A firecracker was thrown vertically upward with a speed of $20 \, \mathrm{m/s}$. One second after the flight began, it exploded into two unequal parts, with their mass ratio being $1:2$. Immediately after the explosion, the smaller fragment flew horizontally with a speed of $16 \, \mathrm{m/s}$. Find the speed (in m/s) of the second fragment immediately after the explosion. Assume the acceleration due to gravity is $10 \, \mathrm{m/s}^2$. | 17\, | 1/8 |
In rectangle $PQRS$, $PQ=8$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$, where $k$, $m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$? | 7 | 3/8 |
The integer 119 is a multiple of which number? | 7 | 4/8 |
Given the function \( f(x)=\sqrt{3} \sin 2x + 2 \cos^2 x + a \), if the minimum value of \( f(x) \) on the interval \(\left[ 0, \frac{\pi}{2} \right] \) is \(-1\), find the value of \( a \). | -1 | 7/8 |
Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for any positive integers $m, n$ the number $$ (f(m))^2+ 2mf(n) + f(n^2) $$ is the square of an integer.
*Proposed by Fedir Yudin* | f(n)=n | 2/8 |
Define the function \( f \) on positive integers such that
\[ f(1) = 1, \quad f(3) = 3 \]
and
\[
\begin{aligned}
f(2n) &= f(n), \\
f(4n+1) &= 2f(2n+1) - f(n), \\
f(4n+3) &= 3f(2n+1) - 2f(n)
\end{aligned}
\]
for every positive integer \( n \).
Determine the number of integers \( n \) satisfying \( 1 \leq n \leq 1988 \) for which \( f(n) = n \). | 92 | 1/8 |
A convex pentagon $P=ABCDE$ is inscribed in a circle of radius $1$ . Find the maximum area of $P$ subject to the condition that the chords $AC$ and $BD$ are perpendicular. | 1 + \frac{3\sqrt{3}}{4} | 4/8 |
A square with an integer side length was cut into 2020 smaller squares. It is known that the areas of 2019 of these squares are 1, while the area of the 2020th square is not equal to 1. Find all possible values that the area of the 2020th square can take. Provide the smallest of these possible area values in the answer. | 112225 | 2/8 |
Find the smallest natural number \( n \) such that in a simple graph with 10 vertices and \( n \) edges that is 2-colored, there always exists a monochromatic triangle or a monochromatic quadrilateral. | 31 | 4/8 |
Given vectors $\overrightarrow{a}, \overrightarrow{b}$ satisfy $\overrightarrow{a}=(1, \sqrt {3}), |\overrightarrow{b}|=1, |\overrightarrow{a}+ \overrightarrow{b}|= \sqrt {3}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______. | \frac{2\pi}{3} | 4/8 |
Given two non-zero vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $| \overrightarrow {a} + \overrightarrow {b} | = | \overrightarrow {a} - \overrightarrow {b} | = \sqrt {2} | \overrightarrow {a} |$, find the angle between vector $\overrightarrow {b}$ and $\overrightarrow {a} + \overrightarrow {b}$. | \frac{\pi}{4} | 4/8 |
Determine all positive integers $n$ for which the equation $$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$ has an integer as a solution. | n=1 | 4/8 |
Before the soccer match between the "North" and "South" teams, five predictions were made:
a) There will be no draw;
b) "South" will concede goals;
c) "North" will win;
d) "North" will not lose;
e) Exactly 3 goals will be scored in the match.
After the match, it was found that exactly three predictions were correct. What was the final score of the match? | 2-1 | 1/8 |
Let $ n > 1, n \in \mathbb{Z}$ and $ B \equal{}\{1,2,\ldots, 2^n\}.$ A subset $ A$ of $ B$ is called weird if it contains exactly one of the distinct elements $ x,y \in B$ such that the sum of $ x$ and $ y$ is a power of two. How many weird subsets does $ B$ have? | 2^{n+1} | 3/8 |
Investigate the stability of the equilibrium point $x=0, y=0$ for the system
$$
\left\{\begin{array}{l}
\frac{d x}{d t}=y^{3}+x^{5} \\
\frac{d y}{d t}=x^{3}+y^{5}
\end{array}\right.
$$ | Unstable | 6/8 |
What is the value of $x + y$ if the sequence $3, ~9, ~15, \ldots, ~x, ~y, ~39$ is an arithmetic sequence? | 60 | 2/8 |
Consider the region $A^{}_{}$ in the complex plane that consists of all points $z^{}_{}$ such that both $\frac{z^{}_{}}{40}$ and $\frac{40^{}_{}}{\overline{z}}$ have real and imaginary parts between $0^{}_{}$ and $1^{}_{}$, inclusive. Find the area of $A.$ | 1200 - 200 \pi | 5/8 |
The diagram shows an octagon consisting of $10$ unit squares. The portion below $\overline{PQ}$ is a unit square and a triangle with base $5$. If $\overline{PQ}$ bisects the area of the octagon, what is the ratio $\dfrac{XQ}{QY}$? | \frac{2}{3} | 1/8 |
Calculate the definite integral:
$$
\int_{0}^{\sqrt{2} / 2} \frac{x^{4} \cdot d x}{\sqrt{\left(1-x^{2}\right)^{3}}}
$$ | \frac{5}{4} - \frac{3\pi}{8} | 3/8 |
From the vertex of the right angle of triangle \(ABC\), the median \(CM\) is drawn. The circle inscribed in triangle \(CAM\) touches \(CM\) at its midpoint. Find the angle \(BAC\). | 60 | 7/8 |
It is known that one of the roots of the equation \(x^{2} - 4a^{2}b^{2}x = 4\) is \(x_{1} = (a^{2} + b^{2})^{2}\). Find \(a^{4} - b^{4}\). | \2 | 1/8 |
Let $S$ be the set of $2\times2$ -matrices over $\mathbb{F}_{p}$ with trace $1$ and determinant $0$ . Determine $|S|$ . | p(p+1) | 4/8 |
Let $n\ge 2$ be a positive integer. There are $n$ real coefficient polynomials $P_1(x),P_2(x),\cdots ,P_n(x)$ which is not all the same, and their leading coefficients are positive. Prove that $$ \deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i) $$ and find when the equality holds. | P_1^n+P_2^n+\cdots+P_n^n-nP_1P_2\cdotsP_n)\ge(n-2)\max_{1\lei\len}(\P_i) | 1/8 |
Let $p(x)$ be a polynomial of degree strictly less than $100$ and such that it does not have $(x^3-x)$ as a factor. If $$ \frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)} $$ for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$ . | 200 | 1/8 |
Using the six digits 0, 1, 2, 3, 4, 5 to form a six-digit number without repeating any digit, how many such numbers are there where the unit digit is less than the ten's digit? | 300 | 7/8 |
What is the minimum number of cells that need to be marked on a chessboard so that
1) none of the marked cells are adjacent (they do not share a side or a corner),
2) adding any one cell to these marked cells would violate condition 1? | 9 | 1/8 |
Using each of the nine digits exactly once, form prime numbers (numbers that are divisible only by 1 and themselves) such that their sum is minimized. | 207 | 1/8 |
A ray of light passing through the point $A = (-3,9,11),$ reflects off the plane $x + y + z = 12$ at $B,$ and then passes through the point $C = (3,5,9).$ Find the point $B.$
[asy]
import three;
size(180);
currentprojection = perspective(6,3,2);
triple A, B, C;
A = (0,-0.5,0.5*1.5);
B = (0,0,0);
C = (0,0.8,0.8*1.5);
draw(surface((-1,-1,0)--(-1,1,0)--(1,1,0)--(1,-1,0)--cycle),paleyellow,nolight);
draw((-1,-1,0)--(-1,1,0)--(1,1,0)--(1,-1,0)--cycle);
draw(A--B--C,Arrow3(6));
label("$A$", A, NW);
label("$B$", B, S);
label("$C$", C, NE);
[/asy] | \left( -\frac{5}{3}, \frac{16}{3}, \frac{25}{3} \right) | 5/8 |
In the plane rectangular coordinate system $xOy$, the equation of the hyperbola $C$ is $x^{2}-y^{2}=1$. Find all real numbers $a$ greater than 1 that satisfy the following requirement: Through the point $(a, 0)$, draw any two mutually perpendicular lines $l_{1}$ and $l_{2}$. If $l_{1}$ intersects the hyperbola $C$ at points $P$ and $Q$, and $l_{2}$ intersects $C$ at points $R$ and $S$, then $|PQ| = |RS|$ always holds. | \sqrt{2} | 4/8 |
Given six cards with the digits $1, 2, 4, 5, 8$ and a comma. Using each card exactly once, various numbers are formed (the comma cannot be at the beginning or at the end of the number). What is the arithmetic mean of all such numbers?
(M. V. Karlukova) | 1234.4321 | 1/8 |
Suppose that $P(x)$ is a monic cubic polynomial with integer roots, and suppose that $\frac{P(a)}{a}$ is an integer for exactly $6$ integer values of $a$ . Suppose furthermore that exactly one of the distinct numbers $\frac{P(1) + P(-1)}{2}$ and $\frac{P(1) - P(-1)}{2}$ is a perfect square. Given that $P(0) > 0$ , find the second-smallest possible value of $P(0).$ *Proposed by Andrew Wu* | 25 | 1/8 |
Given a sphere O with a radius of 2, a cone is inscribed in the sphere O. When the volume of the cone is maximized, find the radius of the sphere inscribed in the cone. | \frac{4(\sqrt{3} - 1)}{3} | 5/8 |
Let \( f(x) \) be an odd function defined on \( \mathbf{R} \), and for \( x \geq 0 \), \( f(x) = x^2 \). If the inequality \( f(x+a) \geq 2 f(x) \) holds for any \( x \in [a, a+2] \), determine the range of the real number \( a \). | [\sqrt{2},+\infty) | 2/8 |
An infinite sequence of positive numbers \(a_{1}, a_{2}, a_{3}, \ldots\) is defined by the rule: \(a_{1}=1\), and \(a_{n+1}^{2}=a_{n}^{2}+\frac{1}{a_{n}}\) for \(n=1, 2, 3, \ldots\). Prove that the sequence \(b_{1}, b_{2}, b_{3}, \ldots\), where \(b_{n}=a_{n+1}-a_{n}\), is decreasing, i.e., \(b_{1} > b_{2} > b_{3} > \ldots\). | b_1>b_2>b_3>\ldots | 1/8 |
There are three boxes of stones. Each hour, Sisyphus moves a stone from one box to another. For each transfer of a stone, he receives from Zeus a number of coins equal to the number of stones in the box from which the stone is drawn minus the number of stones in the recipient box, with the stone Sisyphus just carried not counted. If this number is negative, Sisyphus pays the corresponding amount (and can pay later if he is broke).
After 1000 years, all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment? | 0 | 1/8 |
Are there five consecutive natural numbers such that if they are designated by the letters \(a, b, c, d, e\) in some order, the equality
\[
(a+b)(b+c)(c+d)(d+e)(e+a) = (a+c)(c+e)(e+b)(b+d)(d+a)
\]
holds? | No | 3/8 |
An electronic clock always displays the date as an eight-digit number. For example, January 1, 2011, is displayed as 20110101. What is the last day of 2011 that can be evenly divided by 101? The date is displayed as $\overline{2011 \mathrm{ABCD}}$. What is $\overline{\mathrm{ABCD}}$? | 1221 | 7/8 |
A square pyramid with a base edge of $26 \mathrm{~cm}$ has adjacent triangular faces that form a $120^{\circ}$ angle with each other. How tall is the pyramid? | 13\, | 1/8 |
The solid shown has a square base of side length $s$. The upper edge is parallel to the base and has length $2s$. All other edges have length $s$. Given that $s=6\sqrt{2}$, what is the volume of the solid?
[asy] size(180); import three; pathpen = black+linewidth(0.65); pointpen = black; currentprojection = perspective(30,-20,10); real s = 6 * 2^.5; triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6); draw(A--B--C--D--A--E--D); draw(B--F--C); draw(E--F); label("A",A,W); label("B",B,S); label("C",C,SE); label("D",D,NE); label("E",E,N); label("F",F,N); [/asy] | 288 | 3/8 |
Given an ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$ passes through points $M(2,0)$ and $N(0,1)$.
$(1)$ Find the equation of ellipse $C$ and its eccentricity;
$(2)$ A line $y=kx (k \in \mathbb{R}, k \neq 0)$ intersects ellipse $C$ at points $A$ and $B$, point $D$ is a moving point on ellipse $C$, and $|AD| = |BD|$. Does the area of $\triangle ABD$ have a minimum value? If it exists, find the equation of line $AB$; if not, explain why. | \dfrac{8}{5} | 1/8 |
A car departed from point A to point B, and with some delay, a second car followed. When the first car had traveled half the distance, the second car had traveled $26 \frac{1}{4}$ km. When the second car had traveled half the distance, the first car had traveled $31 \frac{1}{5}$ km. After overtaking the first car, the second car reached point B, immediately turned back, and after driving 2 km, met the first car. Find the distance between points A and B. Give your answer as a number without units. | 58 | 1/8 |
Two spheres touch the plane of triangle \(ABC\) at points \(A\) and \(B\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 9, and the distance between their centers is \(\sqrt{305}\). The center of a third sphere with a radius of 7 is at point \(C\), and it externally touches each of the first two spheres. Find the radius of the circumcircle of triangle \(ABC\). | 2\sqrt{14} | 4/8 |
In the subtraction shown, $K, L, M$, and $N$ are digits. What is the value of $K+L+M+N$?\n$$\begin{array}{r}6 K 0 L \\ -\quad M 9 N 4 \\ \hline 2011\end{array}$$ | 17 | 2/8 |
A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals?
$\textbf{(A)}\ 8\sqrt{3}\qquad\textbf{(B)}\ 10\sqrt{2}\qquad\textbf{(C)}\ 16\sqrt{3}\qquad\textbf{(D)}\ 20\sqrt{2}\qquad\textbf{(E)}\ 40\sqrt{2}$ | \textbf{(D)}\20\sqrt{2} | 1/8 |
Determine the largest natural number $m$ such that for each non negative real numbers $a_1 \ge a_2 \ge ... \ge a_{2014} \ge 0$ , it is true that $$ \frac{a_1+a_2+...+a_m}{m}\ge \sqrt{\frac{a_1^2+a_2^2+...+a_{2014}^2}{2014}} $$ | 44 | 2/8 |
What is the smallest positive angle \( x \) for which
\[
2^{\sin^2 x} \cdot 4^{\cos^2 x} \cdot 2^{\tan x} = 8
\] | 60 | 1/8 |
Prove that there exists an unique sequence $ \left( c_n \right)_{n\ge 1} $ of real numbers from the interval $ (0,1) $ such that $$ \int_0^1 \frac{dx}{1+x^m} =\frac{1}{1+c_m^m } , $$ for all natural numbers $ m, $ and calculate $ \lim_{k\to\infty } kc_k^k. $ *Radu Pop* | \ln(2) | 1/8 |
A cube with a side length of 1 meter was cut into smaller cubes with a side length of 1 centimeter and arranged in a straight line. What is the length of the resulting line? | 10000 | 7/8 |
Each of the numbers \(1, 2, 3, 4, 5, 6\) is to be placed in the cells of a \(2 \times 3\) table, with one number in each cell. In how many ways can this be done so that in each row and in each column the sum of the numbers is divisible by 3? | 48 | 1/8 |
In a right-angled triangle, let \( s_{a} \) and \( s_{b} \) be the medians to the legs, and \( s_{c} \) be the median to the hypotenuse. Determine the maximum value of the expression \( \frac{s_{a} + s_{b}}{s_{c}} \). | \sqrt{10} | 7/8 |
Find the maximum of the expression
$$
|| \ldots|| x_{1}-x_{2}\left|-x_{3}\right|-\ldots\left|-x_{2023}\right|,
$$
where \( x_{1}, x_{2}, \ldots, x_{2023} \) are distinct natural numbers between 1 and 2023. | 2022 | 1/8 |
On a particular street in Waterloo, there are exactly 14 houses, each numbered with an integer between 500 and 599, inclusive. The 14 house numbers form an arithmetic sequence in which 7 terms are even and 7 terms are odd. One of the houses is numbered 555 and none of the remaining 13 numbers has two equal digits. What is the smallest of the 14 house numbers?
(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 is an arithmetic sequence with four terms.) | 506 | 4/8 |
The arithmetic sequence \( a, a+d, a+2d, a+3d, \ldots, a+(n-1)d \) has the following properties:
- When the first, third, fifth, and so on terms are added, up to and including the last term, the sum is 320.
- When the first, fourth, seventh, and so on, terms are added, up to and including the last term, the sum is 224.
What is the sum of the whole sequence? | 608 | 4/8 |
Given the function $f(x) = x^2 - 2\cos{\theta}x + 1$, where $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{2}\right]$.
(1) When $\theta = \frac{\pi}{3}$, find the maximum and minimum values of $f(x)$.
(2) If $f(x)$ is a monotonous function on $x \in \left[-\frac{\sqrt{3}}{2}, \frac{1}{2}\right]$ and $\theta \in [0, 2\pi)$, find the range of $\theta$.
(3) If $\sin{\alpha}$ and $\cos{\alpha}$ are the two real roots of the equation $f(x) = \frac{1}{4} + \cos{\theta}$, find the value of $\frac{\tan^2{\alpha} + 1}{\tan{\alpha}}$. | \frac{16 + 4\sqrt{11}}{5} | 7/8 |
Is it possible to select 1000 points in a plane so that at least 6000 distances between two of them are equal? | Yes | 2/8 |
For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ For example, $\tau (1)=1$ and $\tau(6) =4.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $a$ denote the number of positive integers $n \leq 2005$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \leq 2005$ with $S(n)$ even. Find $|a-b|.$
| 25 | 7/8 |
Given a cube \( ABCDA_1B_1C_1D_1 \) with edge length \( a \), a point \( M \) is taken on the line \( AA_1 \), and a point \( N \) is taken on the line \( BC \) such that the line \( MN \) intersects the edge \( C_1D_1 \). Find the minimum value of the length \(|MN|\). | 3a | 2/8 |
Determine the largest value of \( x \) for which
\[ \left|x^{2}-4x-39601\right| \geq \left|x^{2}+4x-39601\right| \] | 199 | 7/8 |
Pyramid $OABCD$ has square base $ABCD,$ congruent edges $\overline{OA}, \overline{OB}, \overline{OC},$ and $\overline{OD},$ and $\angle AOB=45^\circ.$ Let $\theta$ be the measure of the dihedral angle formed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m$ and $n$ are integers, find $m+n.$
| 5 | 5/8 |
For an even integer positive integer $n$ Kevin has a tape of length $4 n$ with marks at $-2 n,-2 n+1, \ldots, 2 n-1,2 n$. He then randomly picks $n$ points in the set $-n,-n+1,-n+2, \ldots, n-1, n$, and places a stone on each of these points. We call a stone 'stuck' if it is on $2 n$ or $-2 n$, or either all the points to the right, or all the points to the left, all contain stones. Then, every minute, Kevin shifts the unstuck stones in the following manner: He picks an unstuck stone uniformly at random and then flips a fair coin. If the coin came up heads, he then moves that stone and every stone in the largest contiguous set containing that stone one point to the left. If the coin came up tails, he moves every stone in that set one point right instead. He repeats until all the stones are stuck. Let $p_{k}$ be the probability that at the end of the process there are exactly $k$ stones in the right half. Evaluate $$\frac{p_{n-1}-p_{n-2}+p_{n-3}-\ldots+p_{3}-p_{2}+p_{1}}{p_{n-1}+p_{n-2}+p_{n-3}+\ldots+p_{3}+p_{2}+p_{1}}$$ in terms of $n$. | \frac{1}{n-1} | 1/8 |
The line $y = \frac{5}{3} x - \frac{17}{3}$ is to be parameterized using vectors. Which of the following options are valid parameterizations?
(A) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ 1 \end{pmatrix} + t \begin{pmatrix} -3 \\ -5 \end{pmatrix}$
(B) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 17 \\ 5 \end{pmatrix} + t \begin{pmatrix} 6 \\ 10 \end{pmatrix}$
(C) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ -7/3 \end{pmatrix} + t \begin{pmatrix} 3/5 \\ 1 \end{pmatrix}$
(D) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 14/5 \\ -1 \end{pmatrix} + t \begin{pmatrix} 1 \\ 3/5 \end{pmatrix}$
(E) $\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ -17/3 \end{pmatrix} + t \begin{pmatrix} 15 \\ -25 \end{pmatrix}$
Enter the letters of the correct options, separated by commas. | \text{A,C} | 6/8 |
Let \( x_{1} = 1 \) and \( x_{n+1} = x_{n} + \left\lfloor \frac{x_{n}}{n} \right\rfloor + 2 \) for \( n = 1, 2, 3, \ldots \), where \( \lfloor x \rfloor \) denotes the largest integer not greater than \( x \). Determine \( x_{1997} \). | 23913 | 1/8 |
Doraemon and Nobita are playing the game "rock, paper, scissors." The rules state that the winner of each round receives two dorayakis, while the loser gets none. If there is a tie, each player receives one dorayaki. Nobita knows that Doraemon can only play "rock," but he still wants to share dorayakis with Doraemon. Therefore, he decides to play "scissors" once in every ten rounds and then play "rock" for the remaining rounds. After 20 rounds, all the dorayakis have been distributed, and Nobita has received 30 dorayakis. How many dorayakis did Doraemon receive? | 10 | 1/8 |
Triangle $ABC$ has $AB = 13, BC = 14$, and $AC = 15$. The points $D, E$, and $F$ are the midpoints of $\overline{AB}, \overline{BC}$, and $\overline{AC}$ respectively. Let $X \neq E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA + XB + XC$? | \frac{195}{8} | 7/8 |
Let rightangled $\triangle ABC$ be given with right angle at vertex $C$ . Let $D$ be foot of altitude from $C$ and let $k$ be circle that touches $BD$ at $E$ , $CD$ at $F$ and circumcircle of $\triangle ABC$ at $G$ . $a.)$ Prove that points $A$ , $F$ and $G$ are collinear. $b.)$ Express radius of circle $k$ in terms of sides of $\triangle ABC$ . | \frac{}{2} | 1/8 |
Given six people are arranged in a row from left to right. Only A or B can be placed at the far left, and A cannot be placed at the far right, calculate the total number of different arrangements. | 216 | 3/8 |
A set of positive integers is said to be pilak if it can be partitioned into 2 disjoint subsets \(F\) and \(T\), each with at least 2 elements, such that the elements of \(F\) are consecutive Fibonacci numbers, and the elements of \(T\) are consecutive triangular numbers. Find all positive integers \(n\) such that the set containing all the positive divisors of \(n\) except \(n\) itself is pilak. | 30 | 1/8 |
Given an isosceles triangle \( A B C \) where \( A B = A C \) and \( \angle A B C = 53^\circ \). Point \( K \) is such that \( C \) is the midpoint of \( A K \). Point \( M \) is chosen such that:
- \( B \) and \( M \) are on the same side of the line \( A C \);
- \( K M = A B \);
- the angle \( \angle M A K \) is the largest possible.
What is the measure of the angle \( \angle B A M \) in degrees?
| 44 | 1/8 |
For any three positive integers \( x, y, z \), let
\[
f(x, y, z) = [1+2+3+\cdots+(x+y-2)] - z .
\]
Find all positive integer tuples \( (a, b, c, d) \) such that
\[
f(a, b, c) = f(c, d, a) = 1993.
| (23,42,23,42) | 4/8 |
For which natural number \( n \) does the quantity \(\frac{n^2}{1.001^n}\) reach its maximum value? | 2001 | 5/8 |
Thirty-six hundredths is equal to
(A) 0.36
(B) 360
(C) 3.6
(D) 0.036
(E) 0.0036 | 0.36 | 1/8 |
Two circular tracks $\alpha$ and $\beta$ of the same radius are tangent to each other. A car $A$ travels clockwise on track $\alpha$ and a car $B$ travels counterclockwise on track $\beta$. At the start, cars $A$ and $B$ are on the same line with the center of track $\alpha$, and this line is tangent to track $\beta$. After the start, the cars begin to approach the point of tangency of the tracks. Each car completes one full lap on its track in one hour (and never switches to the other track). For how much time during this hour will the distance between the cars be at least the diameter of each track? | 1/2 | 2/8 |
A leak formed in the hold of a ship. A pump was immediately switched on to remove the water, but it couldn't keep up, and after 10 minutes, the water level rose by 20 cm. Then, a second pump of equal power was turned on, and after 5 minutes, the water level dropped by 10 cm. The leak was then sealed.
How much time will it take for the pumps to remove the remaining water? | 1.25 | 1/8 |
Given a function defined on the set of positive integers as follows:
\[ f(n) = \begin{cases}
n - 3, & \text{if } n \geq 1000 \\
f[f(n + 7)], & \text{if } n < 1000
\end{cases} \]
Find the value of \( f(90) \). | 999 | 2/8 |
Let side $AD$ of convex quadrilateral $ABCD$ be extended through $D$, and let side $BC$ be extended through $C$, to meet in point $E.$ Let $S$ be the degree-sum of angles $CDE$ and $DCE$, and let $S'$ represent the degree-sum of angles $BAD$ and $ABC.$ If $r=S/S'$, then: | 1 | 7/8 |
Convert $115_{10}$ to base 11. Represent $10$ as $A$, if necessary. | \text{A5}_{11} | 1/8 |
Denote $\mathbb{Z}_{>0}=\{1,2,3,...\}$ the set of all positive integers. Determine all functions $f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$ such that, for each positive integer $n$,
$\hspace{1cm}i) \sum_{k=1}^{n}f(k)$ is a perfect square, and
$\vspace{0.1cm}$
$\hspace{1cm}ii) f(n)$ divides $n^3$. | f(n) = n^3 | 1/8 |
Find the distance between the foci of the ellipse
\[\frac{x^2}{36} + \frac{y^2}{16} = 8.\] | 8\sqrt{10} | 7/8 |
The percentage of seventh-grade students participating in the gymnastics section is between 2.9% and 3.1%. Determine the smallest possible number of students in this class. | 33 | 3/8 |
When the five numbers 10000, 1, 10, 100, and 1000 are arranged from largest to smallest, the middle number is
(A) 10000
(B) 1
(C) 10
(D) 100
(E) 1000 | 100 | 1/8 |
If $a+b=1$, find the supremum of $$- \frac {1}{2a}- \frac {2}{b}.$$ | - \frac {9}{2} | 3/8 |
Let \( R \) be a point on the curve such that \( OMRN \) is a square. If \( r \) is the \( x \)-coordinate of \( R \), find the value of \( r \). | 1 | 5/8 |
Find all four-digit numbers that are 9 times larger than their reversed counterparts. | 9801 | 5/8 |
A circle is tangent to the extensions of two sides \(AB\) and \(AD\) of the square \(ABCD\), and the points of tangency cut off segments of length 4 cm from vertex \(A\). From point \(C\), two tangents are drawn to this circle. Find the side length of the square if the angle between the tangents is \(60^\circ\). | 4(\sqrt{2}-1) | 4/8 |
In triangle \(ABC\), \(\angle ABC = 100^\circ\) and \(\angle ACB = 65^\circ\). On \(AB\) there is a point \(M\) such that \(\angle MCB = 55^\circ\), and on \(AC\) there is a point \(N\) such that \(\angle NBC = 80^\circ\). Find \(\angle NMC\). | 25 | 2/8 |
On side AC of triangle ABC with a 120-degree angle at vertex B, points D and E are marked such that AD = AB and CE = CB. A perpendicular DF is dropped from point D to line BE. Find the ratio BD / DF. | 2 | 5/8 |
In the 2013 Zhejiang College Entrance Examination, arrange the six letters A, B, C, D, E, F in a row, with both A and B on the same side of C. How many different arrangements are there? (Answer with a number.) | 480 | 6/8 |
All the squares of a board of $(n+1)\times(n-1)$ squares
are painted with **three colors** such that, for any two different
columns and any two different rows, the 4 squares in their
intersections they don't have all the same color. Find the
greatest possible value of $n$ . | 5 | 1/8 |
All the complex roots of $(z + 1)^4 = 16z^4,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. | \frac{2}{3} | 7/8 |
For the polynomial \(\left(x^{2}-x+1\right)^{100}\), find the sum of the coefficients of the even powers of \(x\). | \frac{1+3^{100}}{2} | 7/8 |
Sierpinski's triangle is formed by taking a triangle, and drawing an upside down triangle inside each upright triangle that appears. A snake sees the fractal, but decides that the triangles need circles inside them. Therefore, she draws a circle inscribed in every upside down triangle she sees (assume that the snake can do an infinite amount of work). If the original triangle had side length $1$ , what is the total area of all the individual circles?
*2015 CCA Math Bonanza Lightning Round #4.4* | \frac{\pi}{12} | 7/8 |
When the integer $ {\left(\sqrt{3} \plus{} 5\right)}^{103} \minus{} {\left(\sqrt{3} \minus{} 5\right)}^{103}$ is divided by 9, what is the remainder? | 1 | 2/8 |
The circles $\omega_{1}$ and $\omega_{2}$ with centers $O_{1}$ and $O_{2}$ respectively intersect at point $B$. The extension of segment $O_{2} B$ beyond point $B$ intersects circle $\omega_{1}$ at point $K$, while the extension of segment $O_{1} B$ beyond point $B$ intersects circle $\omega_{2}$ at point $L$. A line passing through point $B$ and parallel to $K L$ intersects circles $\omega_{1}$ and $\omega_{2}$ again at points $A$ and $C$ respectively. The rays $A K$ and $C L$ intersect at point $N$. Find the angle between lines $O_{1} N$ and $O_{2} B$. | 90 | 2/8 |
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$ | 2 | 1/8 |
On a computer keyboard, the key for the digit 1 is not working. For example, if you try to type the number 1231234, only the number 23234 will actually print.
Sasha tried to type an 8-digit number, but only 202020 was printed. How many 8-digit numbers satisfy this condition? | 28 | 6/8 |
End of preview. Expand
in Data Studio
Overview
Training dataset for Polaris Preview models. The dataset is filtered from DeepScaleR-Preview-Dataset and AReal-boba-Data
Format
Each row in the jsonl file contains:
- problem: The input problem.
- answer: The answer to the problem
- difficulty: The pass rate of the problem estimated by
Deepseek-R1-distill-Qwen-7B
Citation
@misc{Polaris2025,
title = {POLARIS: A Post-Training Recipe for Scaling Reinforcement Learning on Advanced Reasoning Models},
url = {https://hkunlp.github.io/blog/2025/Polaris},
author = {An, Chenxin and Xie, Zhihui and Li, Xiaonan and Li, Lei and Zhang, Jun and Gong, Shansan and Zhong, Ming and Xu, Jingjing and Qiu, Xipeng and Wang, Mingxuan and Kong, Lingpeng}
year = {2025}
}
- Downloads last month
- 114