GRPO training pool — 32 random samples

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?

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?

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).

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

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 $3$?

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)

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.

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 $.

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 $a$?

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}

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?

$9.2 \quad C_{x+1}^{x-2}+2 C_{x-1}^{3}=7(x-1)$.

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.

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}

Calculate the product of one-fourth and one-half. Present the answer in LaTex format: \boxed{Your answer}

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}=\\) \_\_\_\_\_\_.

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 $PBC$.

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)

Given that \(a_n + b_n\sqrt{3} = (2+\sqrt{3})^n\), where \(a_n\) and \(b_n\) are integers, compute \(\lim_{n\rightarrow\infty}\frac{a_n}{b_n}\).

Find the rightmost non-zero digit of the expansion of $20 \times 13!$.

Find the least positive integer which is a multiple of $13$ and all its digits are the same.

1. Let $a, b, c \in \mathbf{R}$. If $$ a+b+c=0, a^{2}+b^{2}+c^{2}=6 \text {, } $$ then $a^{4}+b^{4}+c^{4}=$ $\qquad$ .

Given that $-6 \leq x \leq -3$ and $1 \leq y \leq 5$, what is the largest possible value of $\frac{x+y}{x}$?

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}

Determine the cardinality of the set \( \mathcal{R}[0,1] \) of all Riemann integrable real functions on the interval [0,1].

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

If $m$ and $n$ are odd integers, how many terms in the expansion of $(m+n)^6$ are odd?

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 \( \int_{1}^{100} \frac{f(x)}{x} \, dx \).

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.

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 $S$.

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 \(\lim_{n\rightarrow \infty}\int f_n g \, d\mu = \int f g \, d\mu\).

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.