problem stringlengths 28 7.44k | answer stringlengths 1 60 | difficulty stringclasses 8
values | band stringclasses 3
values | source stringclasses 1
value |
|---|---|---|---|---|
Point \( K \) lies on side \( BC \) of parallelogram \( ABCD \), and point \( M \) lies on its side \( AD \). Segments \( CM \) and \( DK \) intersect at point \( L \), and segments \( AK \) and \( BM \) intersect at point \( N \). Find the maximum value of the ratio of the areas of quadrilateral \( KLMN \) to \( ABCD ... | \frac{1}{4} | 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: `Eric`, `Peter`, `Arnold`, `Alice`
- The mothers' names in different house... | Janelle | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
In a square with a side of 1, there is a broken line (polygonal chain) of length $L$. It is known that each point of the square is at a distance less than $\varepsilon$ from some point of this broken line. Prove that $L \geq \frac{1}{2 \varepsilon} - \frac{\pi \varepsilon}{2}$. | L \geq \frac{1}{2\varepsilon} - \frac{\pi \varepsilon}{2} | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Show that for any integer \( x \), the number
$$
x^{9}-6 x^{7}+9 x^{5}-4 x^{3}
$$
is divisible by 8640. | 8640 | 6/8 | 4-7 | 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, Beverage, Nationality, Pet, Sport.
The attributes have the following possible values:
- Job: dancer, entrepreneur, fisherman, lawyer, security-guard, teacher
- Beverage: cola, fanta, hot-... | lawyer | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Find all pairs of prime numbers \((p, q)\) such that
$$
\left(3 p^{q-1}+1\right) \mid \left(11^{p}+17^{p}\right).
$$ | (3,3) | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
The members of a distinguished committee were choosing a president, and each member gave one vote to one of the $27$ candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least $1$ than the number of votes for that candidate. What is the smallest possible number of members of... | 134 | 5/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
Among the 1000 natural numbers from 1 to 1000, find the number of natural numbers that can be neither divisible by 4 nor by 6.
(A) 416
(B) 584
(C) 625
(D) 667 | 667 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let $\alpha$ and $\beta$ be positive integers such that $$ \frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16} . $$ Find the smallest possible value of $\beta$ . | 23 | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes: Job, Food, Beverage, Movie-Genre, Nationality, Transport.
The attributes have the following possible values:
- Job: engineer, freelancer, pilot, videographer
- Food: apricot, cauliflower, pepp... | malaysian | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Given the function \(f: \mathbf{R} \rightarrow \mathbf{R}\) that satisfies
\[
\begin{array}{l}
f(x f(x)+f(x) f(y)+y-1) \\
= f(x f(x)+x y)+y-1,
\end{array}
\]
find the explicit form of the function \(f(x)\). | f(x)=x | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
1. Find the value of the expression $2 a-\left(\frac{2 a-3}{a+1}-\frac{a+1}{2-2 a}-\frac{a^{2}+3}{2 a^{2-2}}\right) \cdot \frac{a^{3}+1}{a^{2}-a}+\frac{2}{a}$ when $a=1580$. | 2 | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Define the sequence $a_0,a_1,\dots$ inductively by $a_0=1$ , $a_1=\frac{1}{2}$ , and
\[a_{n+1}=\dfrac{n a_n^2}{1+(n+1)a_n}, \quad \forall n \ge 1.\]
Show that the series $\displaystyle \sum_{k=0}^\infty \dfrac{a_{k+1}}{a_k}$ converges and determine its value.
*Proposed by Christophe Debry, KU Leuven, Belgium.* | 1 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
If the line $y=mx+1$ intersects the ellipse $x^2+4y^2=1$ exactly once, then the value of $m^2$ is
$\textbf{(A) }\textstyle\frac{1}{2}\qquad \textbf{(B) }\frac{2}{3}\qquad \textbf{(C) }\frac{3}{4}\qquad \textbf{(D) }\frac{4}{5}\qquad \textbf{(E) }\frac{5}{6}$ | \textbf{(C)}\frac{3}{4} | 1/8 | 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, Food, Beverage, Movie-Genre, Music-Genre, Nationality.
The attributes have the following possible values:
- Job: designer, entrepreneur, freelancer, musician, paramedic, scientist
- Food:... | lemonade | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let ${f : \Bbb{R} \rightarrow \Bbb{R}}$ be a continuous and strictly increasing function for which
\[ \displaystyle f^{-1}\left(\frac{f(x)+f(y)}{2}\right)(f(x)+f(y)) =(x+y)f\left(\frac{x+y}{2}\right) \]
for all ${x,y \in \Bbb{R}} ({f^{-1}}$ denotes the inverse of ${f})$ . Prove that there exist real constants ${a... | f(x)=ax+b | 2/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Let the sequence \( u_{0}, u_{1}, u_{2}, \cdots \) be defined as follows:
$$
\begin{array}{l}
u_{0}=2, u_{1}=\frac{5}{2}, \\
u_{n+1}=u_{n}\left(u_{n-1}^{2}-2\right)-u_{1}, \quad n=1,2, \cdots
\end{array}
$$
Prove that \( \left\lfloor u_{n} \right\rfloor = 2^{\frac{2^{n}-(-1)^{n}}{3}}, \quad n=1,2, \cdots \)
where \( \... | 2^{\frac{2^n-(-1)^n}{3}} | 2/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Four identical pieces, in the shape of right triangles, were arranged in two different ways, as shown in the given figures. The squares $A B C D$ and $E F G H$ have sides respectively equal to $3 \mathrm{~cm}$ and $9 \mathrm{~cm}$. Determine the measure of the side of the square $I J K L$.
$, we have $|x-y| \geqslant \frac{1}{25} x y$. How many numbers can $A$ contain at most? | 9 | 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: `Alice`, `Eric`, `Bob`, `Arnold`, `Carol`, `Peter`
- Each person has a uni... | 1 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let the sequence $(a_{n})_{n\geqslant 1}$ be defined as: $$ a_{n}=\sqrt{A_{n+2}^{1}\sqrt[3]{A_{n+3}^{2}\sqrt[4]{A_{n+4}^{3}\sqrt[5]{A_{n+5}^{4}}}}}, $$ where $A_{m}^{k}$ are defined by $$ A_{m}^{k}=\binom{m}{k}\cdot k!. $$ Prove that $$ a_{n}<\frac{119}{120}\cdot n+\frac{7}{3}. $$ | a_n<\frac{119}{120}\cdotn+\frac{7}{3} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Maria is a regional sales representative for a company that sells electronic gadgets. Each week, she needs to manage the inventory levels and ensure that the delivery schedules are met for her region. This week, she starts with 150 gadgets in her inventory. She receives a shipment of 200 more gadgets on Monday. By Wedn... | 70 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
A pedestrian is walking in a straight line towards a crosswalk at a constant speed of 3.6 km/h. Initially, the pedestrian is 20 meters away from the crosswalk. The length of the crosswalk is 5 meters. How far from the crosswalk will the pedestrian be in half a minute? | 5 | 1/8 | 1-3 | 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: `Bob`, `Peter`, `Arnold`, `Alice`, `Eric`
- Everyone has a unique favorite... | science fiction | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let \( S = \{1, 2, 3, 4\} \). A sequence of \( n \) terms \( a_1, a_2, \cdots, a_n \) has the following property: for any non-empty subset \( B \) of \( S \) (denote the number of elements in \( B \) as \( |B| \)), there exist \( |B| \) adjacent terms in the sequence that exactly form the subset \( B \). Find the minim... | 8 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
The value of \( a \) is chosen such that the number of roots of the first equation \( 4^{x} - 4^{-x} = 2 \cos a x \) is 2007. How many roots does the second equation \( 4^{x} + 4^{-x} = 2 \cos a x + 4 \) have for the same \( a \)? | 4014 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
If $\mathit{B}$ is a point on circle $\mathit{C}$ with center $\mathit{P}$, then the set of all points $\mathit{A}$ in the plane of circle $\mathit{C}$ such that the distance between $\mathit{A}$ and $\mathit{B}$ is less than or equal to the distance between $\mathit{A}$
and any other point on circle $\mathit{C}$ is
$... | \textbf{(B)} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
What is the largest integer $n$ that satisfies $(100^2-99^2)(99^2-98^2)\dots(3^2-2^2)(2^2-1^2)$ is divisible by $3^n$ ? | 49 | 7/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for evert integer $n \ge 1$ , $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$ . How many times $111$ appears on this sequence? | 14 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let $m$ be a positive integer, and let $T$ denote the set of all subsets of $\{1, 2, \dots, m\}$. Call a subset $S$ of $T$ $\delta$-[I]good[/I] if for all $s_1, s_2\in S$, $s_1\neq s_2$, $|\Delta (s_1, s_2)|\ge \delta m$, where $\Delta$ denotes the symmetric difference (the symmetric difference of two sets is the set o... | 2048 | 2/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
A gardener was hired by the King for twenty-six days to perform some work in the garden. The King stipulated that for each day the gardener worked diligently, he would receive three pretzels, but if he shirked his duties, he would not only receive nothing but also owe one pretzel.
At the end of the twenty-six days, i... | 4 | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Mom gave Vasya money for 30 pencils. It turned out that the pencil factory is running a promotion in the store: in exchange for a receipt for the purchase of a set of 20 pencils, $25\%$ of the cost of the set is refunded, and in exchange for a receipt for the purchase of a set of 5 pencils, $10\%$ is refunded. What is ... | 36 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let $\mathcal{F}$ be the family of all nonempty finite subsets of $\mathbb{N} \cup \{0\}.$ Find all real numbers $a$ for which the series $$ \sum_{A \in \mathcal{F}} \frac{1}{\sum_{k \in A}a^k} $$ is convergent. | 2 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
How many ways are there to write $657$ as a sum of powers of two, where each power of two is used at most twice in the sum? For example, $256 + 256 + 128 + 16 + 1$ is a valid sum. | 41 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Consider the second-degree polynomial \(P(x) = 4x^2+12x-3015\). Define the sequence of polynomials
\(P_1(x)=\frac{P(x)}{2016}\) and \(P_{n+1}(x)=\frac{P(P_n(x))}{2016}\) for every integer \(n \geq 1\).
[list='a']
[*]Show that exists a real number \(r\) such that \(P_n(r) < 0\) for every positive integer \(n\).
[*]Fin... | 1008 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Xiaoming and Xiaojun start simultaneously from locations A and B, heading towards each other. If both proceed at their original speeds, they meet after 5 hours. If both increase their speeds by 2 km/h, they meet after 3 hours. The distance between locations A and B is 30 km. | 30 | 2/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
A school library purchased 17 identical books. How much do they cost if they paid more than 11 rubles 30 kopecks, but less than 11 rubles 40 kopecks for 9 of these books? | 2142 | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Half of the yellow flowers are tulips, one third of the blue flowers are daisies, and seven tenths of the flowers are yellow. Find the percentage of flowers that are daisies. | 45\% | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Jelena is a Serbian woman who is an emulationist, meaning she enjoys recreating historical events and lifestyles. She is also a fervent supporter of the POKS political party. During a local festival, she sets up a booth to teach children about Serbian history and culture. Jelena prepares 48 traditional Serbian desserts... | 2 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
A lock with three buttons opens if the buttons are pressed in a predetermined order, one after the other. What is the minimum number of button presses needed to ensure that the lock will open? (The correct sequence of three button presses is not affected by any preceding button presses.)
Translating the problem st... | 9 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
As shown in the diagram, four small plates \( A, B, C, D \) are arranged in a circular shape, with an unspecified number of candies placed on each plate. In each move, it is allowed to take all candies from 1, 3, or 4 plates, or from 2 adjacent plates. What is the maximum number of different possible amounts of candies... | 13 | 7/8 | 4-7 | 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: `Eric`, `Peter`, `Arnold`, `Bob`, `Alice`
- Each person lives in a unique ... | Peter | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Given the function $f(x)=x+\sqrt{1-x}$, determine the minimum value of $f(x)$. | \frac{5}{4} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
The focus of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a>0, b>0)\) is \(2c\). The line \(l\) passes through points \((a, 0)\) and \((0, b)\), and the sum of the distances from point \((1,0)\) to line \(l\) and from point \((-1,0)\) to line \(l\) is not less than \(\frac{4}{5}c\). Determine the range of... | [\frac{\sqrt{5}}{2},\sqrt{5}] | 5/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
Parallelogram $ABCD$ is such that angle $B < 90$ and $AB<BC$ . Points E and F are on the circumference of $\omega$ inscribing triangle ABC, such that tangents to $\omega$ in those points pass through D. If $\angle EDA= \angle{FDC}$ , find $\angle{ABC}$ .
| 60 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Find all functions \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) such that for any \( x, y \in \mathbb{N}^{*} \), \( (f(x))^{2} + y \) is divisible by \( f(y) + x^{2} \). | f(x)=x | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Find all real-valued functions $f$ on the reals such that $f(-x) = -f(x)$ , $f(x+1) = f(x) + 1$ for all $x$ , and $f\left(\dfrac{1}{x}\right) = \dfrac{f(x)}{x^2}$ for $x \not = 0$ . | f(x)=x | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
\( f : \mathbb{R}^2 \rightarrow \mathbb{R}^3 \) (where \( \mathbb{R} \) is the real line) is defined by \( f(x, y) = \left( -\frac{y}{x^2 + 4y^2}, \frac{x}{x^2 + 4y^2}, 0 \right) \). Can we find \( F : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \), such that:
1. If \( F = (F_1, F_2, F_3) \), then \( F_i \) all have continuo... | \text{No} | 1/8 | 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: `Peter`, `Bob`, `Eric`, `Carol`, `Alice`, `Arnold`
- The people keep uniqu... | 6 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Consider the infinite series defined by the following progression:
\[2 + \frac{1}{3} + \frac{1}{9} + \frac{1}{3^2} + \frac{1}{9^2} + \frac{1}{3^3} + \frac{1}{9^3} + \cdots\]
Determine the limit of this series as it extends to infinity.
A) $\frac{1}{3}$
B) $3$
C) $\frac{21}{8}$
D) $2\frac{5}{8}$
E) $2.5$ | \frac{21}{8} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
A cooperative farm can purchase two types of feed mixtures from a neighboring farm to feed its animals. The Type I feed costs $30 per sack and contains 10 kg of component A and 10 kg of component B. The Type II feed costs $50 per sack and contains 10 kg of component A, 20 kg of component B, and 5 kg of component C. It ... | 165 | 2/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Let \( S = \{1, 2, \cdots, 2016\} \). For any non-empty finite sets of real numbers \( A \) and \( B \), find the minimum value of
\[ f = |A \Delta S| + |B \Delta S| + |C \Delta S| \]
where
\[ X \Delta Y = \{a \in X \mid a \notin Y\} \cup \{a \in Y \mid a \notin X\} \]
is the symmetric difference between sets \( X \) a... | 2017 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Find the minimum value of \(\sum_{i=1}^{10} \sum_{j=1}^{10} \sum_{k=1}^{10} |k(x+y-10i)(3x-6y-36j)(19x+95y-95k)|\), where \(x\) and \(y\) are any integers. | 2394000000 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Let \(ABCD\) be a square of side length 5. A circle passing through \(A\) is tangent to segment \(CD\) at \(T\) and meets \(AB\) and \(AD\) again at \(X \neq A\) and \(Y \neq A\), respectively. Given that \(XY = 6\), compute \(AT\). | \sqrt{30} | 7/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
Let $n$ be a given positive integer. Let $\mathbb{N}_+$ denote the set of all positive integers.
Determine the number of all finite lists $(a_1,a_2,\cdots,a_m)$ such that:
[b](1)[/b] $m\in \mathbb{N}_+$ and $a_1,a_2,\cdots,a_m\in \mathbb{N}_+$ and $a_1+a_2+\cdots+a_m=n$.
[b](2)[/b] The number of all pairs of integers ... | \dfrac{2^{n-1} + 2^{\lfloor n/2 \rfloor}}{2} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
## Task A-2.1.
Determine all pairs $(a, b)$ of integers such that the area of the triangle whose vertices are the points where the parabola $y=x^{2}+a x+b$ intersects the coordinate axes is 3. | (1, -2) | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Fill the numbers from $1$ to $25$ into a $5 \times 5$ table. Select the maximum number from each row and the minimum number from each column. A total of 10 numbers are selected in this way. Among these 10 selected numbers, there are at least $\qquad$ distinct numbers. | 9 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
What percentage of a seven-by-seven grid is shaded?
In a seven-by-seven grid, alternate squares are shaded starting with the top left square similar to a checkered pattern. However, an entire row (the fourth row from the top) and an entire column (the fourth column from the left) are left completely unshaded. | 73.47\% | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Given \( x, y \in [0,+\infty) \) and satisfying \( x^{3} + y^{3} + 3xy = 1 \).
Find the maximum value of \( x^{2} y \). | \frac{4}{27} | 5/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
438. Escape across the river. During the flight of the Turkish troops at Treise, a small detachment found itself on the bank of a wide and deep river. Here they found a boat in which two boys were boating. The boat was so small that it could only hold two children or one adult.
How did the officer manage to cross the ... | 1432 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Estimate the population of Nisos in the year 2050. | 2000 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
8. (10 points) A certain exam consists of 7 questions, each of which only concerns the answers to these 7 questions, and the answers can only be one of $1, 2, 3, 4$. It is known that the questions are as follows:
(1) How many questions have the answer 4?
(2) How many questions do not have the answer 2 or 3?
(3) What is... | 16 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\,^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black ... | \textbf{(A)}\\frac{49}{512} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
For any $n\in\mathbb N$ , denote by $a_n$ the sum $2+22+222+\cdots+22\ldots2$ , where the last summand consists of $n$ digits of $2$ . Determine the greatest $n$ for which $a_n$ contains exactly $222$ digits of $2$ . | 222 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let's say a positive integer $ n$ is *atresvido* if the set of its divisors (including 1 and $ n$ ) can be split in in 3 subsets such that the sum of the elements of each is the same. Determine the least number of divisors an atresvido number can have. | 16 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Suppose that \(\left(a_{1}, \ldots, a_{20}\right)\) and \(\left(b_{1}, \ldots, b_{20}\right)\) are two sequences of integers such that the sequence \(\left(a_{1}, \ldots, a_{20}, b_{1}, \ldots, b_{20}\right)\) contains each of the numbers \(1, \ldots, 40\) exactly once. What is the maximum possible value of the sum
\[
... | 5530 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Show that the numbers $\tan \left(\frac{r \pi }{15}\right)$ , where $r$ is a positive integer less than $15$ and relatively prime to $15$ , satisfy
\[x^8 - 92x^6 + 134x^4 - 28x^2 + 1 = 0.\] | x^8-92x^6+134x^4-28x^2+1=0 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let triangle $ABC$ have $\angle BAC = 45^{\circ}$ and circumcircle $\Gamma$. Let $M$ be the intersection of the angle bisector of $\angle BAC$ with $\Gamma$. Let $\Omega$ be the circle tangent to segments $\overline{AB}$ and $\overline{AC}$ and internally tangent to $\Gamma$ at point $T$. Given that $\angle TMA = 45^{\... | 12 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
For the set \( \mathrm{T} = \{1, 2, \ldots, 999\} \), find the maximum number \( \mathrm{k} \) of different subsets \( \mathrm{A}_1, \mathrm{A}_2, \ldots, \mathrm{A}_\mathrm{k} \) such that for any \( \mathrm{i}, \mathrm{j} \) with \( 1 \leq \mathrm{i} < \mathrm{j} \leq \mathrm{k} \), we have \( A_i \cup A_j = \mathrm{... | 1000 | 4/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
The solution of the equation \(7^{x+7} = 8^x\) can be expressed in the form \(x = \log_b 7^7\). The original answer is in the format \(\frac{k}{m}\). Please find the value of \(k + m\). | 15 | 2/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Find the largest real number $\lambda$ such that
\[a^2 + b^2 + c^2 + d^2 \ge ab + \lambda bc + cd\]for all nonnegative real numbers $a,$ $b,$ $c,$ $d.$ | \frac{3}{2} | 6/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
The diagram shows the two squares \( BCDE \) and \( FGHI \) inside the triangle \( ABJ \), where \( E \) is the midpoint of \( AB \) and \( C \) is the midpoint of \( FG \). What is the ratio of the area of the square \( BCDE \) to the area of the triangle \( ABJ \)? | 1/3 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
The intersecting squares from left to right have sides of lengths 12, 9, 7, and 3, respectively. By how much is the sum of the black areas greater than the sum of the gray areas? | 103 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let $O$ be the origin. There exists a scalar $k$ so that for any points $A,$ $B,$ $C,$ and $D$ such that
\[3 \overrightarrow{OA} - 2 \overrightarrow{OB} + 5 \overrightarrow{OC} + k \overrightarrow{OD} = \mathbf{0},\]the four points $A,$ $B,$ $C,$ and $D$ are coplanar. Find $k.$ | -6 | 7/8 | 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: `Peter`, `Eric`, `Carol`, `Arnold`, `Bob`, `Alice`
- Each person has a fav... | gray | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
There are 3 houses, numbered 1 to 3 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`, `Peter`, `Arnold`
- Each person has a unique favorite drink: `tea`... | victorian | 1/8 | 1-3 | 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: `Eric`, `Arnold`, `Alice`, `Bob`, `Peter`
- Each person lives in a unique ... | 5 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
The rectangle \(ABCD\) has side \(AB\) of length \(12 \, \text{cm}\) and side \(BC\) of length \(6 \, \text{cm}\). Point \(S\) is the center of the rectangle, and point \(F\) lies on side \(CD\). The quadrilateral \(BCFS\) has an area that is one third of the area of rectangle \(ABCD\).
Determine the length of segment... | 4\, | 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`, `Arnold`, `Bob`, `Carol`, `Alice`, `Eric`
- People have unique he... | short | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
In a mathematics competition, there are 30 problems. Each correctly solved problem is worth 4 points, a wrong solution results in -1 point. If someone does not attempt a problem, they get 0 points for it. How many different total scores can a contestant achieve? | 145 | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
3. Let's call a natural number special if one of its digits can be replaced by another digit so that all digits in the resulting number are distinct. Numbers in which all digits are already distinct are also considered special. How many special ten-digit numbers exist? (20 points) | 150232320 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Given a sequence $\{a_n\}$ where the first term is 1 and the common difference is 2,
(1) Find the general formula for $\{a_n\}$;
(2) Let $b_n=\frac{1}{a_n \cdot a_{n-1}}$, and the sum of the first n terms of the sequence $\{b_n\}$ is $T_n$. Find the minimum value of $T_n$. | \frac{1}{3} | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Each artist in the creative collective "Patience and Labor" has their own working schedule. Six of them paint one picture every two days, another eight of them paint one picture every three days, and the rest never paint pictures. From September 22 to September 26, they painted a total of 30 pictures. How many pictures... | 4 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Find the functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \), we have \( f(x - f(x - y)) + x = f(x + y) \). | f(x)=x | 2/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Points $X$ and $Y$ are the midpoints of arcs $AB$ and $BC$ of the circumscribed circle of triangle $ABC$ . Point $T$ lies on side $AC$ . It turned out that the bisectors of the angles $ATB$ and $BTC$ pass through points $X$ and $Y$ respectively. What angle $B$ can be in triangle $ABC$ ? | 90 | 4/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
There are 3 houses, numbered 1 to 3 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: `Arnold`, `Peter`, `Eric`
- Everyone has something unique for lunch: `pizz... | 2 | 3/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
8. The function $J(x)$ is defined by:
$$
J(x)=\left\{\begin{array}{ll}
4+x & \text { for } x \leq-2, \\
-x & \text { for }-20 .
\end{array}\right.
$$
How many distinct real solutions has the equation $J(J(J(x)))=0$ ? | 4 | 5/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy \[1-3\sin\theta+5\cos3\theta = 0?\]
$\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8$ | \textbf{(D)}6 | 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`, `Carol`, `Eric`, `Alice`, `Bob`, `Arnold`
- People use unique pho... | oneplus 9 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Let \( I \) be the incenter of a non-equilateral triangle \( ABC \), \( I_A \) be the \(A\)-excenter, \( I_A' \) be the reflection of \( I_A \) in \( BC \), and \( l_A \) be the reflection of line \( AI_A' \) in \( AI \). Define points \( I_B, I_B' \) and line \( l_B \) analogously. Let \( P \) be the intersection poin... | \angleXIY=120 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest $\ell > 0$ such that there exists a square of side $\ell$ contained in a cube with edge $1$ . | \frac{\sqrt{6}}{2} | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
Let $I$ be the incenter of $\triangle ABC$ , and $O$ be the excenter corresponding to $B$ . If $|BI|=12$ , $|IO|=18$ , and $|BC|=15$ , then what is $|AB|$ ? | 24 | 1/8 | 1-3 | POLARIS-Project/Polaris-Dataset-53K |
In a football match between the teams "Zubilo" and "Shaiba," bets on "Zubilo" winning were accepted at 1 to 2 (i.e., if "Zubilo" wins, the bettor receives twice the amount they bet), and bets on "Shaiba" winning were accepted at 1 to 3. Volka was able to place a bet knowing that he would get back exactly the same amoun... | 1:6 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
On an island, there are knights who always tell the truth and liars who always lie. There are 11 players on the island's football team. Player number 1 said, "There are as many knights as liars on our team." Player number 2 said, "The number of knights and liars on our team differs by one," and so on. Player number 11 ... | 0 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
4. (8 points) In the movie "The Monkey King's Return", there is a scene where Monkey King battles mountain demons. Some of the demons are knocked down, and the number of those knocked down is one third more than those standing; After a while, 2 more demons are knocked down, but then 10 demons stand up again. At this po... | 35 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Karl bought five folders from Pay-A-Lot at a cost of $\textdollar 2.50$ each.
Pay-A-Lot had a 20%-off sale the following day. How much could
Karl have saved on the purchase by waiting a day?
$\textbf{(A)}\ \textdollar 1.00 \qquad\textbf{(B)}\ \textdollar 2.00 \qquad\textbf{(C)}\ \textdollar 2.50\qquad\textbf{(D)}\ \te... | \textbf{(C)}\\textdollar2.50 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
A doctor told Mikael to take a pill every 75 minutes. He took his first pill at 11:05. At what time did he take his fourth pill? | 14:50 | 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: `Carol`, `Bob`, `Eric`, `Alice`, `Peter`, `Arnold`
- People have unique he... | 1 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
Two ascetics live on the top of a vertical cliff of height \( h \) and at a distance from a neighboring village \( m \) times greater. One ascetic descends the cliff and then walks directly to the village. The other ascetic ascends to a certain height \( x \) and then flies directly to the village. If both of them trav... | \frac{}{+2} | 4/8 | 4-7 | POLARIS-Project/Polaris-Dataset-53K |
Let \(ABCD\) be a convex cyclic quadrilateral with the circumcenter \(U\) where the diagonals are perpendicular to each other. Let \(g\) be the line obtained by reflecting the diagonal \(AC\) over the angle bisector of \(\angle BAD\).
Prove that the point \(U\) lies on the line \(g\). | 0 | 0/8 | 0 | POLARIS-Project/Polaris-Dataset-53K |
End of preview. Expand in Data Studio
Polaris-6K-RLVR
A 6,000-problem subset of POLARIS-Project/Polaris-Dataset-53K, stratified by that dataset's pass-rate difficulty labels and filtered so every item is automatically gradeable.
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.
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. The set is weighted toward the hardest
band by adding 0/8 problems rather than by removing mid-difficulty ones, so the
mid-difficulty portion is not thinned out.
| band | meaning | count | share |
|---|---|---|---|
0 |
0/8 correct — no successful rollout | 4,000 | 66.7% |
1-3 |
1–3/8 correct — low success | 1,500 | 25.0% |
4-7 |
4–7/8 correct — mid success | 500 | 8.3% |
| total | 6,000 | 100% |
Per-difficulty breakdown:
| difficulty | count |
|---|---|
0/8 |
4,000 |
1/8 |
608 |
2/8 |
463 |
3/8 |
429 |
4/8 |
128 |
5/8 |
110 |
6/8 |
101 |
7/8 |
161 |
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.
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/data/build_polaris12k.py --mix "0=4000,1-3=1500,4-7=500" --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 — a larger draw of the same band is a superset of a smaller one.
Citation
Cite the source dataset (POLARIS). This repository only re-stratifies it.
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