Find all functions $ f: \mathbb{Z} \to \mathbb{Z} $ such that $$ f\left(x+f\left(y\right)\right)f\left(y+f\left(x\right)\right)=\left(2x+f\left(y-x\right)\right)\left(2y+f\left(x-y\right)\right) $$ holds for all integers $ x,y $

Find all monotonically increasing or monotonically decreasing functions $f: \mathbb{R}_+\to\mathbb{R}_+$ which satisfy the equation $f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1$ for any two numbers $x$ and $y$ from $\mathbb{R}_+$. Hereby, $\mathbb{R}_+$ is the set of all positive real numbers. [i]Note.[/i] A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically increasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\geq f\left(y\right)$. A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically decreasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\leq f\left(y\right)$.

(a) Positive integers $p,q,r,a$ satisfy $pq=ra^2$, where $r$ is prime and $p,q$ are relatively prime. Prove that one of the numbers $p,q$ is a perfect square. (b) Examine if there exists a prime $p$ such that $p(2^{p+1}-1)$ is a perfect square.

Find the smallest real number $p(\leq 1)$ that satisfies the following condition. (Condition) For real numbers $x_1, x_2, \dots, x_{2024}, y_1, y_2, \dots, y_{2024}$, if [list] [*] $0 \leq x_1 \leq x_2 \leq \dots \leq x_{2024} \leq 1$, [*] $0 \leq y_1 \leq y_2 \leq \dots \leq y_{2024} \leq 1$, [*] $\displaystyle \sum_{i=1}^{2024}x_i = \displaystyle \sum_{i=1}^{2024}y_i = 2024p$, [/list] then the inequality $\displaystyle \sum_{i=1}^{2024}x_i(y_{2025-i}-y_{2024-i}) \geq 1 - p$ holds.

In the triangle $ABC$ its incircle with center $I$ touches its sides $BC, CA$ and $AB$ in the points $A_1, B_1, C_1$ respectively. Through $I$ is drawn a line $\ell$. The points $A', B'$ and $C'$ are reflections of $A_1, B_1, C_1$ with respect to the line $\ell$. Prove that the lines $AA', BB'$ and $CC'$ intersects at a common point.

A solid cube has side length $ 3$ inches. A $ 2$-inch by $ 2$-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid? $ \textbf{(A)}\ 7\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 10\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 15$

There are $11$ points equally spaced on a circle. Some of the segments having endpoints among these vertices are drawn and colored in two colors, so that each segment meets at an internal for it point at most one other segment from the same color. What is the greatest number of segments that could be drawn?

$[BD]$ is a median of $\triangle ABC$. $m(\widehat{ABD})=90^\circ$, $|AB|=2$, and $|AC|=6$. $|BC|=?$ $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 3\sqrt2 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4\sqrt2 \qquad\textbf{(E)}\ 2\sqrt6 $

In the simple and connected graph $G$ let $x_i$ be the number of vertices with degree $i$. Let $d>3$ be the biggest degree in the graph $G$. Prove that if : $$x_d \ge x_{d-1} + 2x_{d-2}+... +(d-1)x_1$$ Then there exists a vertex with degree $d$ such that after removing that vertex the graph $G$ is still connected. Proposed by [i]Ali Mirzaie[/i]

Let $k$ be a fixed positive integer. For each $n = 1, 2,...,$ we will call [i]configuration [/i] of order $n$ any set of $kn$ points of the plane, which does not contain $3$ collinear, colored with $k$ given colors, so that there are $n$ points of each color. Determine all positive integers $n$ with the following property: in each configuration of order $n$, it is possible to select three points of each color, such that the $k$ triangles with vertices of the same color that are determined are disjoint in pairs.

A $6$ meter ladder rests against a vertical wall. The midpoint of the ladder is twice as far from the ground as it is from the wall. At what height on the wall does the ladder reach?

Find the greatest possible distance between any two points inside or along the perimeter of an equilateral triangle with side length $2$. [i]Proposed by Alex Li[/i]

If $ n$ runs through all the positive integers, then $ f(n) \equal{} \left \lfloor n \plus{} \sqrt {3n} \plus{} \frac {1}{2} \right \rfloor$ runs through all positive integers skipping the terms of the sequence $ a_n \equal{} \left \lfloor \frac {n^2 \plus{} 2n}{3} \right \rfloor$.

Given any integer $n\geq 3$. A finite series is called $n$-series if it satisfies the following two conditions $1)$ It has at least $3$ terms and each term of it belongs to $\{ 1,2,...,n\}$ $2)$ If series has $m$ terms $a_1,a_2,...,a_m$ then $(a_{k+1}-a_k)(a_{k+2}-a_k)<0$ for all $k=1,2,...,m-2$ How many $n$-series are there $?$

Five pairs of twins are randomly arranged around a circle. Then they perform zero or more swaps, where each swap switches the positions of two adjacent people. They want to reach a state where no one is adjacent to their twin. Compute the expected value of the smallest number of swaps needed to reach such a state.

Let $n$, $d$ be positive integers such that $d>\frac{n}{2}$. Suppose $a_1, a_2,\cdots,a_{d+2}$ is a sequence of integers satisfying $a_{d+1}=a_1$, $a_{d+2}=a_2$, and for all indices $1\le i_1<i_2<\cdots <i_s\le d$, $$a_{i_1}+a_{i_2}+\cdots+a_{i_s}\not\equiv 0\pmod n$$ Prove that there exists $1\le i\le d$ such that $$a_{i+1}\equiv a_i \pmod n \quad \text{or} \quad a_{i+1}\equiv a_i+a_{i+2} \pmod n$$ [i]Proposed by Yeoh Zi Song[/i]

Alice and Bob play the following game. They alternate selecting distinct nonzero digits (from $1$ to $9$) until they have chosen seven such digits, and then consider the resulting seven-digit number by concatenating the digits in the order selected, with the seventh digit appearing last (i.e. $\overline{A_1B_2A_3B_4A_6B_6A_7}$). Alice wins if and only if the resulting number is the last seven decimal digits of some perfect seventh power. Please determine which player has the winning strategy.

Consider polynomial $f(x)={{x}^{2}}-\alpha x+1$ with $\alpha \in \mathbb{R}.$ a) For $\alpha =\frac{\sqrt{15}}{2}$, let write $f(x)$ as the quotient of two polynomials with nonnegative coefficients. b) Find all value of $\alpha $ such that $f(x)$ can be written as the quotient of two polynomials with nonnegative coefficients.

Leo and Paul are at the Berkeley BART station and are racing to San Francisco. Leo is planning to take the line that takes him directly to SF, and because he has terrible BART luck, his train will arrive in some integer number of minutes, with probability $\frac i{210}$ for $1\le i\le20$ at any given minute. Paul will take a second line, whose trains always arrive before Leo’s train, with uniform probability. However, Paul must also make a transfer to a 3rd line, whose trains arrive with uniform probability between $0$ and $10$ minutes after Paul reaches the transfer station. What is the probability that Leo gets to SF before Paul does?

Given a quadrangle $ABCD$. The lengths of all its sides and diagonals are the rational numbers. Let $O$ be the point of its diagonals intersection. Prove that $|AO|$ - the length of the $[AO]$ segment is also rational.

Let $n\in \mathbb{N}$. Prove that the polynom $p(x)=x^{2n}-2x^{2n-1}+3x^{2n-2}-...-2nx+2n+1$ doesn't have real roots.

Prove that the Fibonacci sequence $\{F_n\}^\infty_{n=1}$ defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for all $n \geq 1$ is a divisibility sequence, that is, if $m\mid n$ then $F_m \mid F_n$ for all positive integers $m$ and $n$.

The figures $ F_1$, $ F_2$, $ F_3$, and $ F_4$ shown are the first in a sequence of figures. For $ n\ge3$, $ F_n$ is constructed from $ F_{n \minus{} 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $ F_{n \minus{} 1}$ had on each side of its outside square. For example, figure $ F_3$ has $ 13$ diamonds. How many diamonds are there in figure $ F_{20}$? [asy]unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); path d=(1/2,0)--(0,sqrt(3)/2)--(-1/2,0)--(0,-sqrt(3)/2)--cycle; marker m=marker(scale(5)*d,Fill); path f1=(0,0); path f2=(0,0)--(-1,1)--(1,1)--(1,-1)--(-1,-1); path[] g2=(-1,1)--(-1,-1)--(0,0)^^(1,-1)--(0,0)--(1,1); path f3=f2--(-2,-2)--(-2,0)--(-2,2)--(0,2)--(2,2)--(2,0)--(2,-2)--(0,-2); path[] g3=g2^^(-2,-2)--(0,-2)^^(2,-2)--(1,-1)^^(1,1)--(2,2)^^(-1,1)--(-2,2); path[] f4=f3^^(-3,-3)--(-3,-1)--(-3,1)--(-3,3)--(-1,3)--(1,3)--(3,3)-- (3,1)--(3,-1)--(3,-3)--(1,-3)--(-1,-3); path[] g4=g3^^(-2,-2)--(-3,-3)--(-1,-3)^^(3,-3)--(2,-2)^^(2,2)--(3,3)^^ (-2,2)--(-3,3); draw(f1,m); draw(shift(5,0)*f2,m); draw(shift(5,0)*g2); draw(shift(12,0)*f3,m); draw(shift(12,0)*g3); draw(shift(21,0)*f4,m); draw(shift(21,0)*g4); label("$F_1$",(0,-4)); label("$F_2$",(5,-4)); label("$F_3$",(12,-4)); label("$F_4$",(21,-4));[/asy]$ \textbf{(A)}\ 401 \qquad \textbf{(B)}\ 485 \qquad \textbf{(C)}\ 585 \qquad \textbf{(D)}\ 626 \qquad \textbf{(E)}\ 761$

$A,B,C$ are points in the plane such that $\angle ABC=90^\circ$. Circles with diameters $BA$ and $BC$ meet at $D$. If $BA=20$ and $BC=21$, then the length of segment $BD$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$? [i]Ray Li[/i]

In the octagon below all sides have the length $1$ and all angles are equal. Determine the distance between the corners $A$ and $B$. [img]https://1.bp.blogspot.com/-i6TAFDvcQ8w/XzXCRhnV_kI/AAAAAAAAMVw/rKrQMfPYYJIaCwl8hhdVHdqO4fIn8O7cwCLcBGAsYHQ/s0/2011%2BMogh%2Bp2.png[/img]