Datasets:
problem stringlengths 25 7.44k | answer stringlengths 1 60 | difficulty stringclasses 8
values | band stringclasses 3
values | source stringclasses 1
value |
|---|---|---|---|---|
Let $G, A_{1}, A_{2}, A_{3}, A_{4}, B_{1}, B_{2}, B_{3}, B_{4}, B_{5}$ be ten points on a circle such that $G A_{1} A_{2} A_{3} A_{4}$ is a regular pentagon and $G B_{1} B_{2} B_{3} B_{4} B_{5}$ is a regular hexagon, and $B_{1}$ lies on minor arc $G A_{1}$. Let $B_{5} B_{3}$ intersect $B_{1} A_{2}$ at $G_{1}$, and let ... | 12^{\circ} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
We are given $n$ identical cubes, each of size $1\times 1\times 1$ . We arrange all of these $n$ cubes to produce one or more congruent rectangular solids, and let $B(n)$ be the number of ways to do this.
For example, if $n=12$ , then one arrangement is twelve $1\times1\times1$ cubes, another is one $3\t... | 13 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Lisa drew graphs of all functions of the form \( y = ax + b \), where \( a \) and \( b \) take all natural values from 1 to 100. How many of these graphs pass through the point \((3, 333)\)? | 33 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
To qualify for the competition, wrestler Vladimir must have three matches and win at least two of them consecutively. His opponents are Andrei (A) and Boris (B). Vladimir can choose to schedule the matches in the order ABA or BAB. The probability of Vladimir losing one match to Boris is 0.3, and to Andrei is 0.4. These... | 0.742 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Bob`, `Alice`, `Carol`
- The people keep uniqu... | knitting | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Given three quadratic equations, each with a leading coefficient of 2 and different non-negative discriminants. Can the discriminant of each equation be a root of the other two equations? | No | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Given \( x_{i} \geq 0 \) for \( i = 1, 2, \cdots, n \) and \( \sum_{i=1}^{n} x_{i} = 1 \) with \( n \geq 2 \), find the maximum value of \( \sum_{1 \leq i \leq j \leq n} x_{i} x_{j} (x_{i} + x_{j}) \). | \frac{1}{4} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Three players play tic-tac-toe together. In other words, the three players take turns placing an "A", "B", and "C", respectively, in one of the free spots of a \(3 \times 3\) grid, and the first player to have three of their label in a row, column, or diagonal wins. How many possible final boards are there where the pl... | 148 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
XL OM - II - Task 3
Given is a trihedral angle $ OABC $ with vertex $ O $ and a point $ P $ inside it. Let $ V $ be the volume of the parallelepiped with two vertices at points $ O $ and $ P $, whose three edges are contained in the rays $ OA^{\rightarrow} $, $ OB^{\rightarrow} $, $ OC^{\rightarrow} $. Calculate the m... | \dfrac{9}{2}V | 2/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
For each positive integer $n$, let
$a_n = \frac{(n+9)!}{(n-1)!}$.
Let $k$ denote the smallest positive integer for which the rightmost nonzero digit of $a_k$ is odd. The rightmost nonzero digit of $a_k$ is | 9 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Eric`, `Arnold`, `Bob`, `Peter`
- They all have a unique favorit... | lilies | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Given 2003 sets, each containing exactly 44 elements, and each pair of sets has exactly one element in common. Find the number of elements in the union of these 2003 sets. | 86130 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Angle ABC is a right angle. The diagram shows four quadrilaterals, where three are squares on each side of triangle ABC, and one square is on the hypotenuse. The sum of the areas of all four squares is 500 square centimeters. What is the number of square centimeters in the area of the largest square? | \frac{500}{3} | 6/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
In bridge, a standard 52-card deck is dealt in the usual way to 4 players. By convention, each hand is assigned a number of "points" based on the formula $$4 \times(\# \mathrm{~A} \text { 's })+3 \times(\# \mathrm{~K} \text { 's })+2 \times(\# \mathrm{Q} \text { 's })+1 \times(\# \mathrm{~J} \text { 's })$$ Given that ... | \frac{197}{1820} | 5/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Peter`, `Bob`, `Eric`, `Arnold`, `Carol`
- Each person has a uni... | pizza | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
A trapezium in the plane is a quadrilateral in which a pair of opposite sides are parallel. A trapezium is said to be non-degenerate if it has positive area. Find the number of mutually non-congruent, non-degenerate trapeziums whose sides are four distinct integers from the set $\{5,6,7,8,9,10\}$. | 31 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
A trapezoid is divided into seven strips of equal width. What fraction of the trapezoid's area is shaded? Explain why your answer is correct. | 4/7 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
380 The four planes in space are mutually parallel, and the distance between each adjacent pair of planes is $h$. A regular tetrahedron has its vertices on the four planes. Find the edge length of the regular tetrahedron. | h\sqrt{10} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Given that the function f(x) defined on the set of real numbers ℝ satisfies f(x+1) = 1/2 + √(f(x) - f^2(x)), find the maximum value of f(0) + f(2017). | 1+\frac{\sqrt{2}}{2} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes: Nationality, Food, Movie-Genre, Beverage, Transport.
The attributes have the following possible values:
- Nationality: german, italian, japanese, mexican, spanish, thai
- Food: cauliflower, cr... | tea | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Eric`, `Bob`, `Arnold`, `Carol`, `Alice`
- Each person has a uni... | 1 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
The number of games won by five basketball teams is shown in a bar chart. The teams' names are not displayed. The following clues provide information about the teams:
1. The Hawks won more games than the Falcons.
2. The Warriors won more games than the Knights, but fewer games than the Royals.
3. The Knights won more ... | 33 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
1. In a $3 \times 4$ table, 12 numbers are arranged such that all seven sums of these numbers in the rows and columns of the table are distinct. What is the maximum number of numbers in this table that can be zero? | 8 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Let $L$ be the intersection point of the lines $AP$ and $CM$, and $S$ be the intersection point of the lines $AN$ and $CQ$. Prove that $LS \| PQ$. | LS\parallelPQ | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Peter`, `Arnold`, `Eric`
- People own unique car models: `honda ... | honda civic | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Inside an equilateral triangle with an area of 1, place five arbitrary points. Prove that within this equilateral triangle, it is always possible to form three smaller equilateral triangles that cover these five points. The sides of these three smaller equilateral triangles are parallel to the sides of the original tri... | 0.64 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
A person starting with $$64$ and making $6$ bets, wins three times and loses three times,
the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss.
If each wager is for half the money remaining at the time of the bet, then the final result is:
$\textbf{(A)}\text{ a loss of... | \textbf{(C)}of37 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Eric`, `Carol`, `Alice`, `Arnold`, `Peter`, `Bob`
- People have unique fa... | oneplus 9 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let $b,m,n\in\mathbb{N}$ with $b>1$ and $m\not=n$ . Suppose that $b^{m}-1$ and $b^{n}-1$ have the same set of prime divisors. Show that $b+1$ must be a power of $2$ . | 1 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
A subset \( X \) of the set of "two-digit" numbers \( 00, 01, \ldots, 98, 99 \) is such that in any infinite sequence of digits, there will be two consecutive digits forming a number from \( X \). What is the smallest number of elements that can be in \( X \)? | 55 | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
An isosceles right triangle is removed from each corner of a square piece of paper to form a rectangle. If $AB = 15$ units in the new configuration, what is the combined area of the four removed triangles? | 112.5 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Arnold`, `Eric`, `Peter`
- They all have a unique favorite flowe... | watermelon | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Determine the maximum number of planes in three-dimensional space such that there exist six points with the following conditions:
i) Each plane contains at least four of the points.
ii) Any four points do not lie on the same line. | 6 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
A convex polygon is given, in which no two sides are parallel. For each of its sides, consider the angle at which it is visible from the vertex most distant from the line containing that side. Prove that the sum of all such angles is $180^{\circ}$. | 180 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
When $10^{95} - 95 - 2$ is expressed as a single whole number, calculate the sum of the digits. | 840 | 7/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
In each cell of a $2019 \times 2019$ square, both diagonals are drawn. Is there a closed path consisting of these diagonals that does not pass through any diagonal more than once and visits all the cells of the square (i.e., contains at least one diagonal from each cell)? | \text{No} | 6/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
Find all real solutions \((x, y)\) to the simultaneous equations:
\[
\frac{1}{x} - \frac{1}{2y} = 2y^4 - 2x^4
\]
\[
\frac{1}{x} + \frac{1}{2y} = (3x^2 + y^2)(x^2 + 3y^2)
\] | (\frac{3^{\frac{1}{5}}+1}{2},\frac{3^{\frac{1}{5}}-1}{2}) | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
The acronym AMC is shown in the rectangular grid below with grid lines spaced $1$ unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC$?$
[asy] import olympiad; unitsize(25); for (int i = 0; i < 3; ++i) { for (int j = 0; j < 9; ++j) { pair A = (j,i); } } for (int i = 0; i... | \textbf{(C)}13+4\sqrt{2} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
The bases of a trapezoid are 8 and 2. The angles adjacent to the larger base are each $45^{\circ}$. Find the volume of the solid formed by rotating the trapezoid about its larger base. | 36\pi | 6/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
8. Let the set $T=\left\{x_{1}, x_{2}, \cdots, x_{10}\right\}$ have five-element subsets such that any two elements of $T$ appear in at most two subsets. The maximum number of such subsets is $\qquad$ . | 18 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
A square with a side length of 36 cm was cut into three rectangles in such a way that the areas of all three rectangles are equal and any two rectangles have a common section of the boundary. What is the total length (in cm) of the made cuts? | 60 | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$ , $A_2=(a_2,a_2^2)$ , $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$ . Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$ -axis at an acute angle of $\theta$ . The maximum possible value fo... | 503 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
8th Putnam 1948 Problem A4 Let D be a disk radius r. Given (x, y) ∈ D, and R > 0, let a(x, y, R) be the length of the arc of the circle center (x, y), radius R, which is outside D. Evaluate lim R→0 R -2 ∫ D a(x, y, R) dx dy. Solution | 4\pi r | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Given that all four roots of the equation $2x^4 + mx^2 + 8 = 0$ are integers, then $m = \ $, and the polynomial $2x^4 + mx^2 + 8$ can be factored into $\ $. | -10 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let \( S \) be a set of 2017 distinct points in the plane. Let \( R \) be the radius of the smallest circle containing all points in \( S \) on either the interior or boundary. Also, let \( D \) be the longest distance between two of the points in \( S \). Let \( a, b \) be real numbers such that \( a \leq \frac{D}{R} ... | (\sqrt{3},2) | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
In triangle \( \triangle ABC \), the internal angles satisfy \(\angle C = 3(\angle A - \angle B)\). \(D\) and \(E\) are the feet of the altitudes from \(A\) to \(AC\) and \(AB\) respectively. The orthocenter of \( \triangle ABC \) is \( H \), and \(M\) is the midpoint of \( AB \). \(P\) is a point on the circumcircle o... | M,R,Q | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Given that \( a > 0 \), if \( f(g(h(a))) = 17 \), where \( f(x) = x^2 + 5 \), \( g(x) = x^2 - 3 \), and \( h(x) = 2x + 1 \), what is the value of \( a \)? | \frac{-1 + \sqrt{3 + 2\sqrt{3}}}{2} | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Let $\{a_k\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$, and in general,
\[a_k=\begin{cases}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is odd,}\\(0.\underbrace{20101\cdots 01011}_{k... | 1341 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Consider a \(7 \times 7\) grid of squares. Let \( f:\{1, 2, 3, 4, 5, 6, 7\} \rightarrow \{1, 2, 3, 4, 5, 6, 7\} \) be a function; in other words, \( f(1), f(2), \ldots, f(7) \) are each (not necessarily distinct) integers from 1 to 7. In the top row of the grid, the numbers from 1 to 7 are written in order; in every ot... | 1470 | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes: Job, Hobby, Nationality, Pet, Transport.
The attributes have the following possible values:
- Job: designer, entrepreneur, freelancer, manager, musician, videographer
- Hobby: camping, photogr... | goat | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Given the hyperbola
$$
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad (a>0, b>0),
$$
with left and right foci $F_{1}$ and $F_{2}$ respectively. A perpendicular is drawn from $F_{2}$ to the $x$-axis, intersecting the hyperbola at points $A$ and $B$. The radius of the inscribed circle of $\triangle A B F_{1}$ is $a$. D... | \frac{1+\sqrt{5}}{2} | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
$A B C$ is an acute triangle with incircle $\omega$. $\omega$ is tangent to sides $\overline{B C}, \overline{C A}$, and $\overline{A B}$ at $D, E$, and $F$ respectively. $P$ is a point on the altitude from $A$ such that $\Gamma$, the circle with diameter $\overline{A P}$, is tangent to $\omega$. $\Gamma$ intersects $\o... | \frac{675}{4} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Point \( C \) is located 12 km downstream from point \( B \). A fisherman set out in a rowboat from point \( A \), which is upstream from point \( B \), and reached \( C \) in 4 hours. For the return trip, he took 6 hours. On another occasion, the fisherman used a motorized boat, tripling his relative speed in the wate... | 1 | 4/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
From the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, four different numbers are selected, denoted as $a$, $b$, $c$, $d$ respectively. If the parity of $a+b$ is the same as the parity of $c+d$, then the total number of ways to select $a$, $b$, $c$, $d$ is ______ (provide the answer in numerical form). | 912 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let \( A \) be a set of \( m \) real numbers with the sum of its elements equal to \( a \); \( B \) be a set of \( n \) real numbers with the sum of its elements equal to \( b \). Define
\[
C = \{\boldsymbol{x} \mid \boldsymbol{x} = (s, t), s \in A, t \in B\}.
\]
Let \(\boldsymbol{x}\) and \(\boldsymbol{y}\) each ra... | n^2^2+^2b^2 | 2/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
There was a solution in a glass, consisting of 99% water. The glass with the solution was weighed and found to weigh 500 grams. After some water evaporated, the water content became 98%. How much will the glass with the resulting solution weigh if the empty glass weighs 300 grams? | 400\, | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
For \( x \) a real number, let \( f(x) = 0 \) if \( x < 1 \) and \( f(x) = 2x - 2 \) if \( x \geq 1 \). How many solutions are there to the equation
\[
f(f(f(f(x)))) = x ?
\] | 2 | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
The points $E$ on the side $BC$ and $F$ on the side $AD$ of a convex quadrilateral $ABCD$ are such that $BE=2EC$ and $AF=2FD$. There is a circle with radius $r$ centered on segment $AE$, which is tangent to the sides $AB$, $BC$, and $CD$. There is another circle with the same radius $r$ centered on segment $BF$, which... | 8r^2 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
$\textbf{Anna's Vintage Postcards}$
Anna organizes the postcards in her collection by country and by the decade in which they were issued. The prices she paid for them at a collectibles shop were: Italy and Germany, $7$ cents each; Japan $5$ cents each; and India $6$ cents each. (Italy and Germany are European countri... | \$1.41 | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
On the coordinate plane, the graphs of three reduced quadratic polynomials intersect the y-axis at the points $-15,-6,-27$ respectively. For each of the polynomials, the coefficient at $x$ is a natural number, and the larger root is a prime number. Find the sum of all roots of these polynomials. | -9 | 7/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Arnold`, `Peter`, `Eric`
- Each person lives in a unique style o... | Peter | 2/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Given a positive real sequence \(a_{1}, a_{2}, \cdots\) such that for any \(k \in \mathbf{Z}_{+}\), it satisfies
\[ a_{k+1} \geqslant \frac{k a_{k}}{a_{k}^{2}+k-1} \text{.} \]
Prove: For every positive integer \(n \geq 2\), we have
\[ \sum_{i=1}^{n} a_{i} \geq n . \] | \sum_{i=1}^{n}a_{i}\gen | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? | \frac{3}{16} | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
There are 5 people standing in a line numbered 1 through 5 in a left to right order.
Each person has the following attributes: Job, Hobby, Movie-Genre, Transport.
The attributes have the following possible values:
- Job: accountant, musician, nurse, paramedic, social-worker
- Hobby: camping, dancing, hiking, rock-clim... | paramedic | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Example 4 In the border desert area, patrol vehicles travel 200 kilometers per day, and each vehicle can carry enough gasoline to travel for 14 days. There are 5 patrol vehicles that set out from base $A$ simultaneously, complete their tasks, and then return along the same route to the base. To allow 3 of them to patro... | 1800 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
How many positive odd integers greater than 1 and less than $150$ are square-free? | 59 | 2/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Given the function defined on the interval $0 \le x < 1$:
\[
f_{1}(x) = \begin{cases}
1, & \text{if } 0.5 \le x < 0.6 \\
0, & \text{otherwise}
\end{cases}
\]
determine the elements of the sequence of functions $f_{n}(x)$ step by step using the following procedure. Where $f_{n-1}(x) > 0$, let $f_{n}(x) = f_{n-1}... | \frac{1}{10-9q} | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Let $M$ be the intersection of diagonals of the convex quadrilateral $ABCD$, where $m(\widehat{AMB})=60^\circ$. Let the points $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of the triangles $ABM$, $BCM$, $CDM$, $DAM$, respectively. Calculate the ratio of the area of quadrilateral $ABCD$ to the area of quadrilateral $... | 3/2 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Consider a triangle \(ABC\), where \(AB = 20\), \(BC = 25\), and \(CA = 17\). \(P\) is a point on the plane. What is the minimum value of \(2 \times PA + 3 \times PB + 5 \times PC\)? | 109 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Eleven best football teams played one match against each other. Each team scored 1 goal in the first match, 2 goals in the second match, up to 10 goals in the tenth match. What is the maximum number of drawn matches that could have occurred? | 50 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Place the numbers $0, 1, 2, 3, 4, 5, 6, 7$ on the eight vertices of a cube (each vertex having one number, with each number used exactly once) in such a way that the sum of the two numbers at the ends of each edge is always a prime number. What is the maximum sum of the four numbers on one face of the cube? | 18 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Eric`, `Arnold`, `Peter`
- Everyone has a unique favorite cigar:... | Fred | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let \( L \) denote the minimum value of the quotient of a 3-digit number formed by three distinct digits divided by the sum of its digits. Determine \( \lfloor 10L \rfloor \). | 105 | 2/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Let \(I\) be the set of points \((x, y)\) in the Cartesian plane such that
\[ x > \left( \frac{y^4}{9} + 2015 \right)^{1/4} \]
Let \(f(r)\) denote the area of the intersection of \(I\) and the disk \(x^2 + y^2 \leq r^2\) of radius \(r > 0\) centered at the origin \((0,0)\). Determine the minimum possible real number \... | \frac{\pi}{3} | 2/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Gari took a 6-item multiple choice test with 3 choices per item, labeled \( A \), \( B \), and \( C \). He remembered that he never answered three consecutive \( A \)'s, he never answered three consecutive \( B \)'s, and he did not leave any item blank. How many possible sets of answers could Gari have had? | 569 | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Let \( P_{1} P_{2} \ldots P_{8} \) be a convex octagon. An integer \( i \) is chosen uniformly at random from 1 to 7, inclusive. For each vertex of the octagon, the line between that vertex and the vertex \( i \) vertices to the right is painted red. What is the expected number of times two red lines intersect at a poi... | \frac{54}{7} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Eric`, `Carol`, `Bob`, `Arnold`, `Peter`
- People have unique fa... | 5 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
A group of tourists started their hike from a campsite. Fifteen minutes later, a tourist named Ivan remembered that he had forgotten a flashlight at the campsite and went back to get it, walking at a speed faster than that of the main group. After retrieving the flashlight, he started catching up with the group at the ... | 1.2 | 4/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
The graph of the function $f(x)=\sin(2x+\varphi)$ is translated to the right by $\frac{\pi}{12}$ units and then becomes symmetric about the $y$-axis. Determine the maximum value of the function $f(x)$ in the interval $\left[0, \frac{\pi}{4}\right]$. | \frac{1}{2} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
4. In space, there are 4 points $A, B, C, D$, satisfying $A B=B C=C D$. If $\angle A B C=\angle B C D=\angle C D A=36^{\circ}$, then the angle formed by line $A C$ and line $B D$ is $\qquad$ | 36^\circ | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
In triangle \( \triangle ABC \), points \( P_1 \) and \( P_2 \) lie on side \( AB \) such that point \( P_2 \) is on segment \( BP_1 \) and \( AP_1 = BP_2 \). Points \( Q_1 \) and \( Q_2 \) lie on side \( BC \) such that point \( Q_2 \) is on segment \( BQ_1 \) and \( BQ_1 = CQ_2 \). Denote the intersection of line seg... | \angleP_1RS=\angleQ_1RM | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
The quadratic function \(y=f(x)\) has a leading coefficient of 1. The coefficients of the linear term and the constant term are integers. Given that \(f(f(x))=0\) has four distinct real roots, and these roots can be arranged in an arithmetic sequence, find the function \(f(x)\) such that the sum of its coefficients is ... | x^2+22x+105 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
A group of 6 students decided to make [i]study groups[/i] and [i]service activity groups[/i] according to the following principle:
Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members.
Supposing there ... | 8 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Convert $115_{10}$ to base 11. Represent $10$ as $A$, if necessary. | \text{A5}_{11} | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Given that the function $f(x)$ is defined on $\mathbb{R}$ and is not identically zero, and for any real numbers $x$, $y$, it satisfies: $f(2)=2$, $f(xy)=xf(y)+yf(x)$, $a_{n}= \dfrac {f(2^{n})}{2^{n}}(n\in\mathbb{N}^{*})$, $b_{n}= \dfrac {f(2^{n})}{n}(n\in\mathbb{N}^{*})$, consider the following statements:
$(1)f(1)=1... | (2)(3)(4) | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
The surface area of a sphere is \(400\pi \text{ cm}^2\). Calculate the volume of the sphere in cubic centimeters and express your answer in terms of \(\pi\). | \frac{4000}{3}\pi \text{ cm}^3 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Snow White has a row of 101 plaster dwarfs in her garden, arranged by weight from heaviest to lightest, with the weight difference between each pair of adjacent dwarfs being the same. Once, Snow White weighed the dwarfs and discovered that the first, heaviest dwarf weighs exactly $5 \mathrm{~kg}$. Snow White was most s... | 2.5 | 7/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
Example 4 Let $A=\left\{x \mid x^{2}+(p+2) x+1=0, x \in \mathbf{R}\right\}$, if $A \varsubsetneqq \mathbf{R}^{-}$, find the range of real number $p$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | (-4, +\infty) | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
A ladder is formed by removing some consecutive unit squares of a $10 \times 10$ chessboard such that for each $k$-th row ($k \in \{1,2,\dots, 10\}$), the leftmost $k-1$ unit squares are removed. Find the number of rectangles formed by the composition of unit squares that the ladder has. | 715 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
How many diagonals in a regular 32-sided polygon are not parallel to any of its sides? | 240 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Let \( n \geqslant 2 \) be an integer. Consider all circular arrangements of the numbers \( 0, 1, \ldots, n \); the \( n+1 \) rotations of an arrangement are considered to be equal. A circular arrangement is called beautiful if, for any four distinct numbers \( 0 \leqslant a, b, c, d \leqslant n \) with \( a+c = b+d \)... | N+1 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let \( n \) be a positive integer. When \( n > 100 \), what are the first two decimal places of the fractional part of \( \sqrt{n^2 + 3n + 1} \)? | 49 | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Carol`, `Arnold`, `Bob`, `Eric`, `Peter`, `Alice`
- People use unique pho... | 5 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and $5^n<2^m<2^{m+2}<5^{n+1}$? | 279 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let $ABC$ be a triangle, and let $L, M, N$ be the midpoints of $[BC], [CA],$ and $[AB]$ respectively. Let $P$ be a point on $[AB]$ and $R$ the point symmetric to $P$ about $N$. Let $Q$ be a point on $[BC]$ and $S$ the point symmetric to $Q$ about $L$. Show that if the lines $(PS)$ and $(QR)$ are perpendicular, their in... | Tliesonthecircumcircleof\triangleLMN | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Peter`, `Eric`, `Arnold`, `Alice`
- Each person has a unique birthday mon... | Peter | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
From a given point, a tangent and a secant are drawn to a given circle. The length of the tangent is $18 \mathrm{~cm}$, and the part of the secant that lies inside the circle is $27 \mathrm{~cm}$. How long is the other part of the secant? | 9\, | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:
- Each person has a unique name: `Alice`, `Peter`, `Bob`, `Eric`, `Arnold`
- Each person has a unique level... | samsung galaxy s21 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
There are 6 people standing in a line numbered 1 through 6 in a left to right order.
Each person has the following attributes: Job, Food, Beverage, Transport.
The attributes have the following possible values:
- Job: analyst, lawyer, librarian, paramedic, pilot, videographer
- Food: grapefruit, onion, pineapple, plum,... | librarian | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Two circles have centers at (1,3) and (4,1) respectively. A line is tangent to the first circle at point (4,6) and to the second circle at point (7,4). Find the slope of the tangent line at these points. | -1 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
End of preview. Expand in Data Studio
Polaris-12K-RLVR
A 12,000-problem RL (RLVR) training set stratified from POLARIS-Project/Polaris-Dataset-53K by that dataset's pass-rate difficulty labels.
Every problem has a short, machine-checkable answer — items whose gold answer is
empty, longer than 60 characters, or multi-line (proof-style) are dropped, so the
whole set is gradeable under a \boxed{} + math_verify policy.
Why 12K and not the full 53K
Reinforcement learning here depends on revisiting the same problem across epochs — anything that keys off per-problem history is never exercised if each problem is seen once. With 256 problems per optimizer step (grad-accum 64 × group 16 × 4 GPUs ÷ 16 rollouts), a 700-step run gives:
| set | visits per problem @700 steps |
|---|---|
| full 53K | 3.4 |
| 12K (this set) | 14.9 |
| 9K | 19.9 |
12K still yields 8.5 visits per problem at 400 steps, so the revisit margin is comfortable while the set stays large enough to be non-trivial.
Composition
Bands are the pass-rate labels from the source dataset — k/8, i.e. how many of
8 rollouts were correct. Lower is harder.
What the labels are, precisely. The source dataset states the difficulty is
"the pass rate of the problem estimated by Deepseek-R1-distill-Qwen-7B". They
are therefore not measured with the model you are likely to train, and they were
produced under POLARIS's own generation budget — that work trained Qwen3-4B with a
40K response length, raised to 52K during RL, and recommends ≥64K at
evaluation. Under a shorter budget the same label is substantially harder than it
looks. Treat the bands as a relative ordering, not an absolute difficulty.
| band | meaning | count | share |
|---|---|---|---|
0 |
0/8 correct — no successful rollout | 8,000 | 66.7% |
1-3 |
1–3/8 correct — low success | 3,000 | 25.0% |
4-7 |
4–7/8 correct — mid success | 1,000 | 8.3% |
| total | 12,000 | 100% |
Per-difficulty breakdown:
| difficulty | count |
|---|---|
0/8 |
8,000 |
1/8 |
1,220 |
2/8 |
914 |
3/8 |
866 |
4/8 |
245 |
5/8 |
237 |
6/8 |
191 |
7/8 |
327 |
The set is deliberately skewed toward the hardest band (0/8): that region is
where reward sparsity actually bites, and it is the intended stress test. The
shape is 3,000 per band plus an extra 3,000 of the 0/8 band — the skew is
made by adding hard problems rather than by removing mid-difficulty ones, so the
mid-difficulty baseline signal is not thinned out in the process.
Note also that, for the reasons in the label caveat above, the effective
zero-signal fraction under a short generation budget is higher than the 50% label
share — the 1-3/8 band partly behaves like the zero band.
Fields
| field | description |
|---|---|
problem |
problem statement |
answer |
gold answer (short, gradeable) |
difficulty |
source pass-rate label, k/8 (Deepseek-R1-distill-Qwen-7B) |
band |
0 / 1-3 / 4-7 |
source |
provenance string |
Reproduce
python scripts/build_polaris12k.py --mix "0=8000,1-3=3000,4-7=1000" --seed 42
Band samples are drawn by shuffling each band's pool under a per-band fixed seed and taking a prefix, so changing one band's count leaves the other bands' samples untouched.
Citation
Cite the source dataset (POLARIS). This repository only re-stratifies it.
- Downloads last month
- 44