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| <title>GRPO pool — 32 random problems</title> | |
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| <h1>GRPO training pool — 32 random samples</h1> | |
| <div class="sub">semantic-decontaminated (bge-large ≥0.85 + LCS + lang + fig, vs all 5 benchmarks) + teacher-resolve gold-confirm (Qwen-1.5B k=4). Colors = difficulty tier.</div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#1</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">DAPO</span></div> | |
| <div class="prob">There are $4$ pairs of men and women, and all $8$ people are arranged in a row such that in each pair, the woman is somewhere to the left of the man. How many such arrangements are there?</div> | |
| <div class="gold"><b>gold:</b> <code>2520</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#2</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">Skywork</span></div> | |
| <div class="prob">Robyn has 4 tasks to do and Sasha has 14 tasks to do. How many of Sasha's tasks should Robyn do in order for them to have the same number of tasks?</div> | |
| <div class="gold"><b>gold:</b> <code>5</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#3</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">DAPO</span></div> | |
| <div class="prob">Find the least positive integer $n$ such that the prime factorizations of $n$, $n + 1$, and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not).</div> | |
| <div class="gold"><b>gold:</b> <code>33</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#4</span> | |
| <span class="tag tier-med">med</span><span class="tag src">OpenR1</span></div> | |
| <div class="prob">To investigate the linear correlation between two variables $x$ and $y$, two students, A and B, independently conducted 10 and 15 experiments, respectively, and used the linear regression method to obtain the regression lines $l_1$ and $l_2$. It is known that in their experiments, the average observed values of variable $x$ were exactly the same, both being $a$; the average observed values of variable $y$ were also exactly the same, both being $b$. Which of the following statements is correct? ( ) | |
| A: The lines $l_1$ and $l_2$ intersect at the point $(a, b)$ | |
| B: The lines $l_1$ and $l_2$ intersect, but the intersection point is not necessarily $(a, b)$ | |
| C: The slopes of lines $l_1$ and $l_2$ are equal, so they must be parallel | |
| D: The lines $l_1$ and $l_2$ must coincide</div> | |
| <div class="gold"><b>gold:</b> <code>A</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#5</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">DAPO</span></div> | |
| <div class="prob">A $\textit{palindrome}$ is a positive integer which reads the same forward and backward, like $12321$ or $4884$. | |
| How many $4$-digit palindromes are divisible by $3$?</div> | |
| <div class="gold"><b>gold:</b> <code>30</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#6</span> | |
| <span class="tag tier-med">med</span><span class="tag src">OpenR1</span></div> | |
| <div class="prob">2. From a three-digit number, the sum of its digits was subtracted. The same operation was performed on the resulting number, and so on, 100 times. Prove that the result will be zero. (6 points)</div> | |
| <div class="gold"><b>gold:</b> <code>0</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#7</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">DAPO</span></div> | |
| <div class="prob">Find the maximal possible number of integers you can choose from the set \( \{1, 2, \ldots, 100\} \) such that no product of any non-empty subset of these numbers is a perfect square.</div> | |
| <div class="gold"><b>gold:</b> <code>25</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#8</span> | |
| <span class="tag tier-easy">easy</span><span class="tag src">Caco</span></div> | |
| <div class="prob">A parabola is given by the equation $ 2y = 3x^2 $. Express this equation in the standard form $ y = ax^2 + b $, and find the value of $ a + b $.</div> | |
| <div class="gold"><b>gold:</b> <code>\frac{3}{2}</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#9</span> | |
| <span class="tag tier-med">med</span><span class="tag src">DeepScaleR</span></div> | |
| <div class="prob">Let $a$, $b$, and $c$ be positive integers with $a \ge b \ge c$ such that | |
| $a^2-b^2-c^2+ab=2011$ and | |
| $a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$. | |
| What is $a$?</div> | |
| <div class="gold"><b>gold:</b> <code>253</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#10</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">Eurus</span></div> | |
| <div class="prob">If one root of the quadratic equation $x^{2}-mx-3=0$ with respect to $x$ is $-2$, calculate the value of $m$. | |
| Present the answer in LaTex format: \boxed{Your answer}</div> | |
| <div class="gold"><b>gold:</b> <code>-\frac{1}{2}</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#11</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">DAPO</span></div> | |
| <div class="prob">We are given five watches which can be wound forward. What is the smallest sum of winding intervals that allows us to set them to the same time, regardless of their initial settings?</div> | |
| <div class="gold"><b>gold:</b> <code>24</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#12</span> | |
| <span class="tag tier-easy">easy</span><span class="tag src">MiroMind</span></div> | |
| <div class="prob">$9.2 \quad C_{x+1}^{x-2}+2 C_{x-1}^{3}=7(x-1)$.</div> | |
| <div class="gold"><b>gold:</b> <code>5</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#13</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">Skywork</span></div> | |
| <div class="prob">Brand Z juice claims, "We offer 30% more juice than Brand W at a price that is 15% less." What is the ratio of the unit price of Brand Z juice to the unit price of Brand W juice? Express your answer as a common fraction.</div> | |
| <div class="gold"><b>gold:</b> <code>\frac{17}{26}</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#14</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">Eurus</span></div> | |
| <div class="prob">Initially, there were natural numbers from 1 to 2021, and they were all white. Then Boris painted every third number blue. After that, Vladimir painted every fifth number red. How many numbers remained white? | |
| Present the answer in LaTex format: \boxed{Your answer}</div> | |
| <div class="gold"><b>gold:</b> <code>1078</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#15</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">Eurus</span></div> | |
| <div class="prob">Calculate the product of one-fourth and one-half. | |
| Present the answer in LaTex format: \boxed{Your answer}</div> | |
| <div class="gold"><b>gold:</b> <code>\frac{1}{8}</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#16</span> | |
| <span class="tag tier-med">med</span><span class="tag src">DeepScaleR</span></div> | |
| <div class="prob">Given \\(f(x)=2x^{5}+3x^{3}-2x^{2}+x-1\\), when calculating the value of the function at \\(x=2\\) using the Horner's method, find \\(v_{3}=\\) \_\_\_\_\_\_.</div> | |
| <div class="gold"><b>gold:</b> <code>20</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#17</span> | |
| <span class="tag tier-med">med</span><span class="tag src">DeepScaleR</span></div> | |
| <div class="prob">In the tetrahedron $P-ABC$, edges $PA$, $AB$, and $AC$ are mutually perpendicular, and $PA = AB = AC$. Points $E$ and $F$ are the midpoints of segments $AB$ and $PC$, respectively. Find the sine of the angle between line $EF$ and plane $PBC$.</div> | |
| <div class="gold"><b>gold:</b> <code>\frac{1}{3}</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#18</span> | |
| <span class="tag tier-med">med</span><span class="tag src">OpenR1</span></div> | |
| <div class="prob">Example $3\left\{a_{n}\right\}(n \geqslant 1)$ is a sequence of complex numbers, where $a_{n}$ is defined as: | |
| $$ | |
| a_{n}=(1+\mathrm{i})\left(1+\frac{\mathrm{i}}{\sqrt{2}}\right) \cdots\left(1+\frac{\mathrm{i}}{\sqrt{n}}\right) . | |
| $$ | |
| Does there exist a natural number $m$ such that | |
| $$ | |
| \sum_{n=1}^{m}\left|a_{n}-a_{n+1}\right|=1990 ? | |
| $$ | |
| (1990-1991 Spanish Mathematical Olympiad)</div> | |
| <div class="gold"><b>gold:</b> <code>1990</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#19</span> | |
| <span class="tag tier-med">med</span><span class="tag src">DeepMath</span></div> | |
| <div class="prob">Given that \(a_n + b_n\sqrt{3} = (2+\sqrt{3})^n\), where \(a_n\) and \(b_n\) are integers, compute \(\lim_{n\rightarrow\infty}\frac{a_n}{b_n}\).</div> | |
| <div class="gold"><b>gold:</b> <code>\sqrt{3}</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#20</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">DAPO</span></div> | |
| <div class="prob">Find the rightmost non-zero digit of the expansion of $20 \times 13!$.</div> | |
| <div class="gold"><b>gold:</b> <code>6</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#21</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">DAPO</span></div> | |
| <div class="prob">Find the least positive integer which is a multiple of $13$ and all its digits are the same.</div> | |
| <div class="gold"><b>gold:</b> <code>111111</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#22</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">Skywork</span></div> | |
| <div class="prob">1. Let $a, b, c \in \mathbf{R}$. If | |
| $$ | |
| a+b+c=0, a^{2}+b^{2}+c^{2}=6 \text {, } | |
| $$ | |
| then $a^{4}+b^{4}+c^{4}=$ $\qquad$ .</div> | |
| <div class="gold"><b>gold:</b> <code>18</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#23</span> | |
| <span class="tag tier-med">med</span><span class="tag src">DeepScaleR</span></div> | |
| <div class="prob">Given that $-6 \leq x \leq -3$ and $1 \leq y \leq 5$, what is the largest possible value of $\frac{x+y}{x}$?</div> | |
| <div class="gold"><b>gold:</b> <code>\frac{1}{6}</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#24</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">Eurus</span></div> | |
| <div class="prob">At the end of a basketball tournament game, each of the seven members of the three participating teams shakes hands with each member of the other two teams, and all of the players shake hands with each of the three referees. How many handshakes occur? | |
| Present the answer in LaTex format: \boxed{Your answer}</div> | |
| <div class="gold"><b>gold:</b> <code>210</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#25</span> | |
| <span class="tag tier-med">med</span><span class="tag src">DeepMath</span></div> | |
| <div class="prob">Determine the cardinality of the set \( \mathcal{R}[0,1] \) of all Riemann integrable real functions on the interval [0,1].</div> | |
| <div class="gold"><b>gold:</b> <code>2^{\mathfrak{c}}</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#26</span> | |
| <span class="tag tier-med">med</span><span class="tag src">OpenR1</span></div> | |
| <div class="prob">When using the method of contradiction to prove the proposition "A triangle has at most one obtuse angle," the negation of the conclusion is ( ) | |
| A: There is no obtuse angle | |
| B: There are two obtuse angles | |
| C: There are three obtuse angles | |
| D: There are at least two obtuse angles</div> | |
| <div class="gold"><b>gold:</b> <code>D</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#27</span> | |
| <span class="tag tier-med">med</span><span class="tag src">DeepScaleR</span></div> | |
| <div class="prob">If $m$ and $n$ are odd integers, how many terms in the expansion of $(m+n)^6$ are odd?</div> | |
| <div class="gold"><b>gold:</b> <code>4</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#28</span> | |
| <span class="tag tier-med">med</span><span class="tag src">DeepMath</span></div> | |
| <div class="prob">Given the function \( f(x) \) defined for all positive real numbers \( x \) such that \( f(x) = f\left(\frac{100}{x}\right) \), and that \( \int_{1}^{10} \frac{f(x)}{x} \, dx = 5 \), find the value of \( \int_{1}^{100} \frac{f(x)}{x} \, dx \).</div> | |
| <div class="gold"><b>gold:</b> <code>10</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#29</span> | |
| <span class="tag tier-med">med</span><span class="tag src">DeepMath</span></div> | |
| <div class="prob">Evaluate the integral \( \int_{C}(z-i) \,dz \) where \( C \) is the parabolic segment defined by \( z(t) = t + it^2 \) for \(-1 \le t \le 1\), by integrating along the straight line from \(-1+i\) to \(1+i\) and applying the Closed Curve Theorem.</div> | |
| <div class="gold"><b>gold:</b> <code>0</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#30</span> | |
| <span class="tag tier-hard">hard</span><span class="tag src">DAPO</span></div> | |
| <div class="prob">Sergei chooses two different natural numbers $a$ and $b$. He writes four numbers in a notebook: $a$, $a+2$, $b$, and $b+2$. He then writes all six pairwise products of the numbers in the notebook on the blackboard. Let $S$ be the number of perfect squares on the blackboard. Find the maximum value of $S$.</div> | |
| <div class="gold"><b>gold:</b> <code>2</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#31</span> | |
| <span class="tag tier-med">med</span><span class="tag src">DeepMath</span></div> | |
| <div class="prob">Let \((X, M, \mu)\) be a \(\sigma\)-finite measure space and let \(1<p < \infty\). Suppose \((f_n)_{n \in \mathbb{N}}\) is a bounded sequence in \(L^{p}(\mu)\) such that \(\lim_{n \rightarrow \infty} \int f_n g \, d\mu\) exists for every \(g \in L^{q}(\mu)\). Find \(f \in L^p(\mu)\) such that \(\lim_{n\rightarrow \infty}\int f_n g \, d\mu = \int f g \, d\mu\).</div> | |
| <div class="gold"><b>gold:</b> <code>f</code></div> | |
| </div> | |
| <div class="card"> | |
| <div class="hd"><span class="n">#32</span> | |
| <span class="tag tier-med">med</span><span class="tag src">DeepMath</span></div> | |
| <div class="prob">Calculate the degree of the map \( S: \mathbb{R}^{m+n} \rightarrow \mathbb{R}^{m+n} \) defined by \( S(x, y) = (y, x) \) using the oriented bases of the tangent spaces.</div> | |
| <div class="gold"><b>gold:</b> <code>(-1)^{mn}</code></div> | |
| </div> | |
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