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source stringclasses 6
values | question stringlengths 10 1.02k | ground_truth stringlengths 1 117 | correct float64 0.13 0.88 |
|---|---|---|---|
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$. | 476 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$, and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$. Find the remainder when $m-n$ is divided by $1000$. | 000 | 0.625 |
Open-Reasoner-Zero/orz_math_57k_collection | Each of the $100$ students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are $42$ students who cannot sing, $65$ students who cannot dance, and $29$ students who cannot act. How many students have two of these talents?
$\te... | 64 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle ABC of side length $3$.
The perimeter of the remaining quadrilateral is
$\text{(A)} \ 6 \qquad \text{(B)} \ 6\frac{1}{2} \qquad \text{(C)} \ 7 \qquad \text{(D)} \ 7\frac{1}{2} \qquad \text{(E)} \ 8$ | 8 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | In base $R_1$ the expanded fraction $F_1$ becomes $.373737\cdots$, and the expanded fraction $F_2$ becomes $.737373\cdots$. In base $R_2$ fraction $F_1$, when expanded, becomes $.252525\cdots$, while the fraction $F_2$ becomes $.525252\cdots$. The sum of $R_1$ and $R_2$, each written in the base ten, is:
$\text{(A) } 2... | 19 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Consider functions $f : [0, 1] \rightarrow \mathbb{R}$ which satisfy
(i)$f(x)\ge0$ for all $x$ in $[0, 1]$,
(ii)$f(1) = 1$,
(iii) $f(x) + f(y) \le f(x + y)$ whenever $x$, $y$, and $x + y$ are all in $[0, 1]$.
Find, with proof, the smallest constant $c$ such that
$f(x) \le cx$
for every function ... | 2 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | What is the value of $[\log_{10}(5\log_{10}100)]^2$?
$\textbf{(A)}\ \log_{10}50 \qquad \textbf{(B)}\ 25\qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 2\qquad \textbf{(E)}\ 1$ | 1 | 0.625 |
Open-Reasoner-Zero/orz_math_57k_collection | If $\log_{k}{x}\cdot \log_{5}{k} = 3$, then $x$ equals:
$\textbf{(A)}\ k^6\qquad \textbf{(B)}\ 5k^3\qquad \textbf{(C)}\ k^3\qquad \textbf{(D)}\ 243\qquad \textbf{(E)}\ 125$ | 125 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | A circle of diameter $1$ is removed from a $2\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region?
[asy] fill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray); draw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1)); fill(circle((1,5/4),1/2),white); draw(circle((1,5/4),1/2),linewidth(1)); ... | 5 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Two permutations $a_1, a_2, \ldots, a_{2010}$ and
$b_1, b_2, \ldots, b_{2010}$ of the numbers $1, 2, \ldots, 2010$
are said to intersect if $a_k = b_k$ for some value of $k$ in the
range $1 \le k\le 2010$. Show that there exist $1006$ permutations
of the numbers $1, 2, \ldots, 2010$ such that any other such
permutatio... | 1006 | 0.5 |
Open-Reasoner-Zero/orz_math_57k_collection | $a_1, a_2, \ldots, a_n$ is an arbitrary sequence of positive integers. A member of the sequence is picked at
random. Its value is $a$. Another member is picked at random, independently of the first. Its value is $b$. Then a third value, $c$. Show that the probability that $a + b +c$ is divisible by $3$ is at least $\f... | \frac{1}{4} | 0.375 |
Open-Reasoner-Zero/orz_math_57k_collection | What is the largest number of towns that can meet the following criteria. Each pair is directly linked by just one of air, bus or train. At least one pair is linked by air, at least one pair by bus and at least one pair by train. No town has an air link, a bus link and a train link. No three towns, $A, B, C$ are such t... | 4 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minim... | 98 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubical box?
$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$ | 6 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | The area of the region bounded by the graph of\[x^2+y^2 = 3|x-y| + 3|x+y|\]is $m+n\pi$, where $m$ and $n$ are integers. What is $m + n$?
$\textbf{(A)} ~18\qquad\textbf{(B)} ~27\qquad\textbf{(C)} ~36\qquad\textbf{(D)} ~45\qquad\textbf{(E)} ~54$ | 54 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Find value of $$ \frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx} $$ if $x$ , $y$ and $z$ are real numbers usch that $xyz=1$ | 1 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $f(x) = x^3 - 3x + b$ and $g(x) = x^2 + bx -3$ , where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x)$ = 0 and $g(x) = 0$ have a common root? | 0 | 0.375 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ABC$ be an acute triangle. $PQRS$ is a rectangle with $P$ on $AB$ , $Q$ and $R$ on $BC$ , and $S$ on $AC$ such that $PQRS$ has the largest area among all rectangles $TUVW$ with $T$ on $AB$ , $U$ and $V$ on $BC$ , and $W$ on $AC$ . If $D$ is the point on $BC$ such that $AD\perp B... | 4 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | $a_1=-1$ , $a_2=2$ , and $a_n=\frac {a_{n-1}}{a_{n-2}}$ for $n\geq 3$ . What is $a_{2006}$ ? $
\textbf{(A)}\ -2
\qquad\textbf{(B)}\ -1
\qquad\textbf{(C)}\ -\frac 12
\qquad\textbf{(D)}\ \frac 12
\qquad\textbf{(E)}\ 2
$ | 2 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the number of integer $n$ from the set $\{2000,2001,...,2010\}$ such that $2^{2n} + 2^n + 5$ is divisible by $7$ (A): $0$ , (B): $1$ , (C): $2$ , (D): $3$ , (E) None of the above. | 4 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!"
Claire asks, "If you picked a digit of your favorite number at random and re... | 13 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | In equilateral triangle $ABC$ , the midpoint of $\overline{BC}$ is $M$ . If the circumcircle of triangle $MAB$ has area $36\pi$ , then find the perimeter of the triangle.
*Proposed by Isabella Grabski* | 36 | 0.5 |
Open-Reasoner-Zero/orz_math_57k_collection | Tatjana imagined a polynomial $P(x)$ with nonnegative integer coefficients. Danica is trying to guess the polynomial. In each step, she chooses an integer $k$ and Tatjana tells her the value of $P(k)$ . Find the smallest number of steps Danica needs in order to find the polynomial Tatjana imagined. | 2 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Ethan Song and Bryan Guo are playing an unfair game of rock-paper-scissors. In any game, Ethan has a 2/5 chance to win, 2/5 chance to tie, and 1/5 chance to lose. How many games is Ethan expected to win before losing?
*2022 CCA Math Bonanza Lightning Round 4.3* | 2 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $A$ and $B$ be distinct positive integers such that each has the same number of positive divisors that 2013 has. Compute the least possible value of $\left| A - B \right|$ . | 1 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime? | 25 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Misha has accepted a job in the mines and will produce one ore each day. At the market, he is able to buy or sell one ore for \ $3, buy or sell bundles of three wheat for \$ 12 each, or $\textit{sell}$ one wheat for one ore. His ultimate goal is to build a city, which requires three ore and two wheat. How many dollar... | 9 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$ . | 4 | 0.5 |
Open-Reasoner-Zero/orz_math_57k_collection | Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:
(a) for any integer $n$ , $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than $187$ .
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(... | 187 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | [help me] Let m and n denote the number of digits in $2^{2007}$ and $5^{2007}$ when expressed in base 10. What is the sum m + n? | 2008 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | For a positive integer $n$ , let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$ . Find the number of fair integers less than $100$ . | 7 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Ten birds land on a $10$ -meter-long wire, each at a random point chosen uniformly along the wire. (That is, if we pick out any $x$ -meter portion of the wire, there is an $\tfrac{x}{10}$ probability that a given bird will land there.) What is the probability that every bird sits more than one meter away from its c... | \frac{1}{10^{10}} | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Consider the sequence: $x_1=19,x_2=95,x_{n+2}=\text{lcm} (x_{n+1},x_n)+x_n$ , for $n>1$ , where $\text{lcm} (a,b)$ means the least common multiple of $a$ and $b$ . Find the greatest common divisor of $x_{1995}$ and $x_{1996}$ . | 19 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Show that any representation of 1 as the sum of distinct reciprocals of numbers drawn from the arithmetic progression $\{2,5,8,11,...\}$ such as given in the following example must have at least eight terms: \[1=\frac{1}{2}+\frac{1}{5}+\frac{1}{8}+\frac{1}{11}+\frac{1}{20}+\frac{1}{41}+\frac{1}{110}+\frac{1}{1640}\] | 8 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Two vector fields $\mathbf{F},\mathbf{G}$ are defined on a three dimensional region $W=\{(x,y,z)\in\mathbb{R}^3 : x^2+y^2\leq 1, |z|\leq 1\}$ . $$ \mathbf{F}(x,y,z) = (\sin xy, \sin yz, 0),\quad \mathbf{G} (x,y,z) = (e^{x^2+y^2+z^2}, \cos xz, 0) $$ Evaluate the following integral.
\[\iiint_{W} (\mathbf{G}\cdot \te... | 0 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | At what smallest $n$ is there a convex $n$ -gon for which the sines of all angles are equal and the lengths of all sides are different? | 5 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many orders can he make exactly five burritos? | 25 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | An 8-by-8 square is divided into 64 unit squares in the usual way. Each unit square is colored black or white. The number of black unit squares is even. We can take two adjacent unit squares (forming a 1-by-2 or 2-by-1 rectangle), and flip their colors: black becomes white and white becomes black. We call this oper... | 32 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$ , $k=1,2,...,7$ . Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$ . What is the most probable sum he can get? | 127 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $V_n=\sqrt{F_n^2+F_{n+2}^2}$ , where $F_n$ is the Fibonacci sequence
( $F_1=F_2=1,F_{n+2}=F_{n+1}+F_{n}$ )
Show that $V_n,V_{n+1},V_{n+2}$ are the sides of a triangle with area $1/2$ | \frac{1}{2} | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Find all natural $ x $ for which $ 3x+1 $ and $ 6x-2 $ are perfect squares, and the number $ 6x^2-1 $ is prime. | 1 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | A number $p$ is $perfect$ if the sum of its divisors, except $p$ is $p$ . Let $f$ be a function such that:
$f(n)=0$ , if n is perfect
$f(n)=0$ , if the last digit of n is 4
$f(a.b)=f(a)+f(b)$
Find $f(1998)$ | 0 | 0.75 |
Open-Reasoner-Zero/orz_math_57k_collection | Determine the range of $w(w + x)(w + y)(w + z)$ , where $x, y, z$ , and $w$ are real numbers such that
\[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\] | 0 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $\mathcal{F}$ be the set of continuous functions $f : [0, 1]\to\mathbb{R}$ satisfying $\max_{0\le x\le 1} |f(x)| = 1$ and let $I : \mathcal{F} \to \mathbb{R}$ ,
\[I(f) = \int_0^1 f(x)\, \text{d}x - f(0) + f(1).\]
a) Show that $I(f) < 3$ , for any $f \in \mathcal{F}$ .
b) Determine $\sup\{I(f) \mid f \... | 3 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose $a,b,c,x,y,z$ are pairwisely different real numbers. How many terms in the following can be $1$ at most: $$ \begin{aligned}
&ax+by+cz,&&&&ax+bz+cy,&&&&ay+bx+cz,
&ay+bz+cx,&&&&az+bx+cy,&&&&az+by+cx?
\end{aligned} $$ | 2 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | The graph of ${(x^2 + y^2 - 1)}^3 = x^2 y^3$ is a heart-shaped curve, shown in the figure below.
[asy]
import graph;
unitsize(10);
real f(real x)
{
return sqrt(cbrt(x^4) - 4 x^2 + 4);
}
real g(real x)
{
return (cbrt(x^2) + f(x))/2;
}
real h(real x)
{
return (cbrt(x^2) - f(x)) ... | 7 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $(a_n)\subset (\frac{1}{2},1)$ . Define the sequence $x_0=0,\displaystyle x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}$ . Is this sequence convergent? If yes find the limit. | 1 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Two sequences $\{a_i\}$ and $\{b_i\}$ are defined as follows: $\{ a_i \} = 0, 3, 8, \dots, n^2 - 1, \dots$ and $\{ b_i \} = 2, 5, 10, \dots, n^2 + 1, \dots $ . If both sequences are defined with $i$ ranging across the natural numbers, how many numbers belong to both sequences?
*Proposed by Isabella Grabski* | 0 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | In $10\times 10$ square we choose $n$ cells. In every chosen cell we draw one arrow from the angle to opposite angle. It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or
the distance between their ends is at least 2. What is the maximum possible value of $... | 50 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ABCD$ be a square of side length $4$ . Points $E$ and $F$ are chosen on sides $BC$ and $DA$ , respectively, such that $EF = 5$ . Find the sum of the minimum and maximum possible areas of trapezoid $BEDF$ .
*Proposed by Andrew Wu* | 16 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the minimum positive value of $ 1*2*3*4*...*2020*2021*2022$ where you can replace $*$ as $+$ or $-$ | 1 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ABCD$ be a cyclic quadrilateral, and suppose that $BC = CD = 2$ . Let $I$ be the incenter of triangle $ABD$ . If $AI = 2$ as well, find the minimum value of the length of diagonal $BD$ . | 2\sqrt{3} | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3
\cos4 & \cos5 & \cos 6
\cos7 & \cos8 & \cos 9\end{vmatrix}.$ The argument of $ \co... | 0 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Suppose $\{ x_n \}_{n\geq 1}$ is a sequence of positive real numbers such that $x_1 \geq x_2 \geq x_3 \ldots \geq x_n \ldots$ , and for all $n$ \[ \frac{x_1}{1} + \frac{x_4}{2} + \frac{x_9}{3} + \ldots + \frac{x_{n^2}}{n} \leq 1 . \] Show that for all $k$ \[ \frac{x_1}{1} + \frac{x_2}{2} +\ldots + \frac{x_k}{k}... | 3 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | In square $ABCD$ , $\overline{AC}$ and $\overline{BD}$ meet at point $E$ .
Point $F$ is on $\overline{CD}$ and $\angle CAF = \angle FAD$ .
If $\overline{AF}$ meets $\overline{ED}$ at point $G$ , and if $\overline{EG} = 24$ cm, then find the length of $\overline{CF}$ . | 48 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Points $A$ , $B$ , $C$ , and $D$ lie on a circle. Let $AC$ and $BD$ intersect at point $E$ inside the circle. If $[ABE]\cdot[CDE]=36$ , what is the value of $[ADE]\cdot[BCE]$ ? (Given a triangle $\triangle ABC$ , $[ABC]$ denotes its area.) | 36 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$ , $AB=16$ , $CD=12$ , and $BC<AD$ . A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$ . | 13 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the maximum number of planes in the space, such there are $ 6$ points, that satisfy to the following conditions:
**1.**Each plane contains at least $ 4$ of them
**2.**No four points are collinear. | 6 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | What is the smallest perfect square larger than $1$ with a perfect square number of positive integer factors?
*Ray Li* | 36 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant $X$ , let $t(X)$ be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true: $1)$ For any two friends... | 2 | 0.375 |
Open-Reasoner-Zero/orz_math_57k_collection | In an acute scalene triangle $ABC$ , points $D,E,F$ lie on sides $BC, CA, AB$ , respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$ . Altitudes $AD, BE, CF$ meet at orthocenter $H$ . Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$ . Lines $DP$ and $QH$ i... | 1 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | On a blackboard the product $log_{( )}[ ]\times\dots\times log_{( )}[ ]$ is written (there are 50 logarithms in the product). Donald has $100$ cards: $[2], [3],\dots, [51]$ and $(52),\dots,(101)$ . He is replacing each $()$ with some card of form $(x)$ and each $[]$ with some card of form $[y]$ . Find th... | 0 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$ , where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals. | 49 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Twenty-six people gather in a house. Alicia is friends with only one person, Bruno is friends with two people, Carlos is a friend of three, Daniel is four, Elías is five, and so following each person is friend of a person more than the previous person, until reaching Yvonne, the person number twenty-five, who is a frie... | 13 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the... | 100 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | $ P(x)$ is a quadratic trinomial. What maximum number of terms equal to the sum of the two preceding terms can occur in the sequence $ P(1)$ , $ P(2)$ , $ P(3)$ , $ \dots?$
*Proposed by A. Golovanov* | 2 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$ .
Let $n$ be a positive integer, $n> 1$ . Find the smallest possible number of terms with a non-zero coefficient in a simple $n$ -th degree polynomial with all values at integer places are divisible ... | 2 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other? | 24 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Find all positive integers $x$ such that $2x+1$ is a perfect square but none of the integers $2x+2, 2x+3, \ldots, 3x+2$ are perfect squares. | 4 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | The temperatures $ f^\circ \text{F}$ and $ c^\circ \text{C}$ are equal when $ f \equal{} \frac {9}{5}c \plus{} 32$ . What temperature is the same in both $ ^\circ \text{F}$ and $ ^\circ \text{C}$ ? | -40 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Given triangle $ ABC$ of area 1. Let $ BM$ be the perpendicular from $ B$ to the bisector of angle $ C$ . Determine the area of triangle $ AMC$ . | \frac{1}{2} | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $(a_n)$ be defined by $a_1=a_2=1$ and $a_n=a_{n-1}+a_{n-2}$ for $n>2$ . Compute the sum $\frac{a_1}2+\frac{a_2}{2^2}+\frac{a_3}{2^3}+\ldots$ . | 2 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$ .
(1) Find $ I_1,\ I_2$ .
(2) Find $ \lim_{n\to\infty} I_n$ . | 1 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | (F.Nilov) Given right triangle $ ABC$ with hypothenuse $ AC$ and $ \angle A \equal{} 50^{\circ}$ . Points $ K$ and $ L$ on the cathetus $ BC$ are such that $ \angle KAC \equal{} \angle LAB \equal{} 10^{\circ}$ . Determine the ratio $ CK/LB$ . | 2 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | A set of $8$ problems was prepared for an examination. Each student was given $3$ of them. No two students received more than one common problem. What is the largest possible number of students? | 8 | 0.375 |
Open-Reasoner-Zero/orz_math_57k_collection | For integers $a, b$ , call the lattice point with coordinates $(a,b)$ **basic** if $gcd(a,b)=1$ . A graph takes the basic points as vertices and the edges are drawn in such way: There is an edge between $(a_1,b_1)$ and $(a_2,b_2)$ if and only if $2a_1=2a_2\in \{b_1-b_2, b_2-b_1\}$ or $2b_1=2b_2\in\{a_1-a_2, ... | 1 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible? | 144 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | On a table there are $100$ red and $k$ white buckets for which all of them are initially empty. In each move, a red and a white bucket is selected and an equal amount of water is added to both of them. After some number of moves, there is no empty bucket and for every pair of buckets that are selected together at l... | 100 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $M$ be a finite subset of the plane such that for any two different points $A,B\in M$ there is a point $C\in M$ such that $ABC$ is equilateral. What is the maximal number of points in $M?$ | 3 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$ . | 35 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Determine the number of real roots of the equation ${x^8 -x^7 + 2x^6- 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2}= 0}$ | 0 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Alice is performing a magic trick. She has a standard deck of 52 cards, which she may order beforehand. She invites a volunteer to pick an integer \(0\le n\le 52\), and cuts the deck into a pile with the top \(n\) cards and a pile with the remaining \(52-n\). She then gives both piles to the volunteer, who riffles them... | 26 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Let $P$ be a regular $2006$ -gon. A diagonal is called *good* if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$ . The sides of $P$ are also called *good*.
Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a commo... | 1003 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Some people know each other in a group of people, where "knowing" is a symmetric relation. For a person, we say that it is $social$ if it knows at least $20$ other persons and at least $2$ of those $20$ know each other. For a person, we say that it is $shy$ if it doesn't know at least $20$ other persons and... | 40 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | In a tetrahedral $ABCD$ , given that $\angle ADB=\angle BDC =\angle CDA=\frac{\pi}{3}$ , $AD=BD=3$ , and $CD=2$ . Find the radius of the circumsphere of $ABCD$ . | \sqrt{3} | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Define an operation $\Diamond$ as $ a \Diamond b = 12a - 10b.$ Compute the value of $((((20 \Diamond 22) \Diamond 22) \Diamond 22) \Diamond22).$ | 20 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | The Fibonacci numbers are defined by $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n>2$ . It is well-known that the sum of any $10$ consecutive Fibonacci numbers is divisible by $11$ . Determine the smallest integer $N$ so that the sum of any $N$ consecutive Fibonacci numbers is divisible by $12$ . | 24 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | If $f(x, y) = 3x^2 + 3xy + 1$ and $f(a, b) + 1 = f(b, a) = 42$ , then determine $|a + b|$ . | 3 \sqrt{3} | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | At the round table, $10$ people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) . It is clear thath I have at least one knight and at least one liar.
What is the largest number of those sitting at the table can say: ''Both of my neighbors are knights ''... | 9 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy\equal{}1$ and both branches of the hyperbola $ xy\equal{}\minus{}1.$ (A set $ S$ in the plane is called *convex* if for any two points in $ S$ the line segment connecting them is contained in $ S.$ ) | 4 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | There are three boxes of stones. Sisyphus moves stones one by one between the boxes. Whenever he moves a stone, Zeus gives him the number of coins that is equal to the difference between the number of stones in the box the stone was put in, and that in the box the stone was taken from (the moved stone does not count). ... | 0 | 0.375 |
Open-Reasoner-Zero/orz_math_57k_collection | Consider the set $ S_n$ of all the $ 2^n$ numbers of the type $ 2\pm \sqrt{2 \pm \sqrt {2 \pm ...}},$ where number $ 2$ appears $ n\plus{}1$ times.
$ (a)$ Show that all members of $ S_n$ are real.
$ (b)$ Find the product $ P_n$ of the elements of $ S_n$ . | 2 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | In the convex pentagon $ABCDE$ : $\angle A = \angle C = 90^o$ , $AB = AE, BC = CD, AC = 1$ . Find the area of the pentagon. | \frac{1}{2} | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | In a regular hexagon $ABCDEF$ of side length $8$ and center $K$ , points $W$ and $U$ are chosen on $\overline{AB}$ and $\overline{CD}$ respectively such that $\overline{KW} = 7$ and $\angle WKU = 120^{\circ}$ . Find the area of pentagon $WBCUK$ .
*Proposed by Bradley Guo* | 32\sqrt{3} | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | The equation $ ax^3\plus{}bx^2\plus{}cx\plus{}d\equal{}0$ has three distinct solutions. How many distinct solutions does the following equation have:
$ 4(ax^3\plus{}bx^2\plus{}cx\plus{}d)(3ax\plus{}b)\equal{}(3ax^2\plus{}2bx\plus{}c)^2?$ | 2 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | A cake has a shape of triangle with sides $19,20$ and $21$ . It is allowed to cut it it with a line into two pieces and put them on a round plate such that pieces don't overlap each other and don't stick out of the plate. What is the minimal diameter of the plate? | 21 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Find the remainder when $(1^2+1)(2^2+1)(3^2+1)\dots(42^2+1)$ is divided by $43$ . Your answer should be an integer between $0$ and $42$ . | 4 | 0.125 |
Open-Reasoner-Zero/orz_math_57k_collection | Isosceles triangle $\triangle{ABC}$ has $\angle{ABC}=\angle{ACB}=72^\circ$ and $BC=1$ . If the angle bisector of $\angle{ABC}$ meets $AC$ at $D$ , what is the positive difference between the perimeters of $\triangle{ABD}$ and $\triangle{BCD}$ ?
*2019 CCA Math Bonanza Tiebreaker Round #2* | 1 | 0.25 |
Open-Reasoner-Zero/orz_math_57k_collection | Joshua likes to play with numbers and patterns. Joshua's favorite number is $6$ because it is the units digit of his birth year, $1996$ . Part of the reason Joshua likes the number $6$ so much is that the powers of $6$ all have the same units digit as they grow from $6^1$ :
\begin{align*}6^1&=6,6^2&=36,6^3&=2... | 6 | 0.5 |
Open-Reasoner-Zero/orz_math_57k_collection | The number $123454321$ is written on a blackboard. Evan walks by and erases some (but not all) of the digits, and notices that the resulting number (when spaces are removed) is divisible by $9$ . What is the fewest number of digits he could have erased?
*Ray Li* | 2 | 0.25 |
End of preview. Expand in Data Studio
Math RLVR
A compact math RLVR dataset derived from
Open-Reasoner-Zero/orz_math_57k_collection,
Open-Reasoner-Zero/orz_math_72k_collection_extended,
BytedTsinghua-SIA/DAPO-Math-17k,
agentica-org/DeepScaleR-Preview-Dataset,
nvidia/AceReason-Math,
and
SynthLabsAI/Big-Math-RL-Verified.
Eight responses were sampled from llama3-1.2b-think-step-10000; records with
1–7 correct responses were retained. Each row contains source, question,
ground_truth, and correct, where correct is the fraction of the eight
responses accepted by the math verifier. Usage remains subject to the source
datasets' licensing terms.
Dataset statistics
| Metric | Value |
|---|---|
| Final records | 89,513 |
| Parquet file size | 6,800,175 bytes |
| Sources represented | 6 |
| Correct-response range | 1/8–7/8 |
| Correct responses | Records |
|---|---|
| 1/8 | 39,889 |
| 2/8 | 16,773 |
| 3/8 | 9,517 |
| 4/8 | 6,938 |
| 5/8 | 5,699 |
| 6/8 | 5,245 |
| 7/8 | 5,452 |
| Total | 89,513 |
Source composition
| Source | Records |
|---|---|
| SynthLabsAI/Big-Math-RL-Verified | 57,941 |
| Open-Reasoner-Zero/orz_math_57k_collection | 9,574 |
| Open-Reasoner-Zero/orz_math_72k_collection_extended | 8,797 |
| nvidia/AceReason-Math | 6,200 |
| agentica-org/DeepScaleR-Preview-Dataset | 5,602 |
| BytedTsinghua-SIA/DAPO-Math-17k | 1,399 |
| Total | 89,513 |
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