Given $n \geq 2$ reals $x_1 , x_2 , \dots , x_n.$ Show that $$\prod_{1\leq i < j \leq n} (x_i - x_j)^2 \leq \prod_{i=0}^{n-1} \left(\sum_{j=1}^{n} x_j^{2i}\right)$$ and find all the $(x_1 , x_2 , \dots , x_n)$ where the equality holds.

Let $A, B, C$ and $D$ be points on the circle $\omega$ such that $ABCD$ is a convex quadrilateral. Suppose that $AB$ and $CD$ intersect at a point $E$ such that $A$ is between $B$ and $E$ and that $BD$ and $AC$ intersect at a point $F$. Let $X \ne D$ be the point on $\omega$ such that $DX$ and $EF$ are parallel. Let $Y$ be the reflection of $D$ through $EF$ and suppose that $Y$ is inside the circle $\omega$. Show that $A, X$, and $Y$ are collinear.

Given a triangle $ABC$. $D$ is a moving point on the edge $BC$. Point $E$ and Point $F$ are on the edge $AB$ and $AC$, respectively, such that $BE=CD$ and $CF=BD$. The circumcircle of $\triangle BDE$ and $\triangle CDF$ intersects at another point $P$ other than $D$. Prove that there exists a fixed point $Q$, such that the length of $QP$ is constant.

Ten pirates find a chest filled with golden and silver coins. There are twice as many silver coins in the chest as there are golden. They divide the golden coins in such a way that the difference of the numbers of coins given to any two of the pirates is not divisible by $10.$ Prove that they cannot divide the silver coins in the same way.

For a positive integer $n$ let $a_n$ denote the last digit of $n^{(n^n)}$. Prove that the sequence $(a_n)$ is periodic and determine the length of the minimal period.

Suppose you are given that for some positive integer $n$, $1! + 2! + \ldots + n!$ is a perfect square. Find the sum of all possible values of $n$.

There are $2022$ stones on a table. At the start of the game, Teacher Tseng will choose a positive integer $m$ and let Ming and LTF play a game. LTF is the first to move, and he can remove at most $m$ stones on his round. Then the two people take turns removing stone, each round they must remove at least one stone, and they cannot remove more than twice the amount of stones the last person removed. The player unable to move loses. Find the smallest positive integer $m$ such that LTF has a winning strategy. [i]Proposed by ltf0501[/i]

While Steve and LeRoy are fishing $ 1$ mile from shore, their boat springs a leak, and water comes in at a constant rate of $ 10$ gallons per minute. The boat will sink if it takes in more than $ 30$ gallons of water. Steve starts rowing toward the shore at a constant rate of $ 4$ miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 10$

How many $2$ digit number $n=ab$ ($a$ and $b$ are digits) have the property that $$n=a+b+a\times b$$ (A)$20$ (B)$15$ (C)$9$ (D)$8$

Alice and Bob are put in charge of building a bridge with their respective teams. With both teams' combined effort, the team can be finished in $6$ days. In reality, Alice's team works alone for the first $3$ days, and then, they decide to take a break. Bob's team takes over from there and works for another $4$ days. As a result, $60\%$ of the bridge is successfully constructed. How many days would it take for Alice's team alone to finish building the bridge completely from the start?

Let $ a,b,c $ be pairwise coprime positive integers. Find all positive integer values of $$ \frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} $$

Calculate the determinant of the $ n\times n $ complex matrix $ \left(a_j^i\right)_{1\le j\le n}^{1\le i\le n} $ defined by $$ a_j^i=\left\{\begin{matrix} 1+x^2,\quad i=j\\x,\quad |i-j|=1\\0,\quad |i-j|\ge 2\end{matrix}\right. , $$ where $ n $ is a natural number greater than $ 2. $

Prove that there exists a $k_0\in\mathbb{N}$ such that for every $k\in\mathbb{N},k>k_0$, there exists a finite number of lines in the plane not all parallel to one of them, that divide the plane exactly in $k$ regions. Find $k_0$.

Let $S$ be the set of all rational numbers that can be expressed as a repeating decimal in the form $0.\overline{abcd},$ where at least one of the digits $a, b, c, $ or $d$ is nonzero. Let $N$ be the number of distinct numerators when numbers in $S$ are written as fractions in lowest terms. For example, both $4$ and $410$ are counted among the distinct numerators for numbers in $S$ because $0.\overline{3636} = \frac{4}{11}$ and $0.\overline{1230} = \frac{410}{3333}.$ Find the remainder when $N$ is divided by $1000.$

Let $D$ be the point on the base $BC$ of an isosceles $\vartriangle ABC$ triangle such that $\frac{\left| BD \right|}{\left| DC \right|}=\text{ }2$, and let $P$ be the point on the segment $\left[ AD \right]$ such that $\angle BAC=\angle BPD$. Prove that $\angle DPC=\frac{1}{2}\angle BAC$.

Prove that there do not exist eleven primes, all less than $20000$, which form an arithmetic progression.

Let $p$ be an even non-negative continous function with $\int _{\mathbb{R}} p(x) dx =1$ and let $n$ be a positive integer. Let $\xi_1,\xi_2,\xi_3 \dots ,\xi_n$ be independent identically distributed random variables with density function $p$ . Define \begin{align*} X_{0} & = 0 \\ X_{1} & = X_0+ \xi_1 \\ & \vdotswithin{ = }\notag \\ X_{n} & = X_{n-1} + \xi_n \end{align*} Prove that the probability that all random variables $X_1,X_2 \dots X_{n-1}$ lie between $X_0$ and $X_n$ is $\frac{1}{n}$. [i]Proposed by Fedor Petrov (St.Petersburg State University).[/i]

The squares $ABCD$ and $AXYZ$ are given. It turns out that $CDXY$ is a cyclic quadrilateral inscribed in the circle $\Omega$, and the points $A, B$ and $Z{}$ lie inside this circle. Prove that either $AB = AX$ or $AC\perp{}XY$.

For two real numbers $ a$, $ b$, with $ ab\neq 1$, define the $ \ast$ operation by \[ a\ast b=\frac{a+b-2ab}{1-ab}.\] Start with a list of $ n\geq 2$ real numbers whose entries $ x$ all satisfy $ 0<x<1$. Select any two numbers $ a$ and $ b$ in the list; remove them and put the number $ a\ast b$ at the end of the list, thereby reducing its length by one. Repeat this procedure until a single number remains. $ a.$ Prove that this single number is the same regardless of the choice of pair at each stage. $ b.$ Suppose that the condition on the numbers $ x$ is weakened to $ 0<x\leq 1$. What happens if the list contains exactly one $ 1$?

The set $ S$ consists of $ n > 2$ points in the plane. The set $ P$ consists of $ m$ lines in the plane such that every line in $ P$ is an axis of symmetry for $ S$. Prove that $ m\leq n$, and determine when equality holds.

Oleksii wrote some \( 2n \) (\( n > 1 \)) consecutive positive integers on the board. After that, he grouped these numbers into pairs in some way, and within each pair, he multiplied the two numbers together. He then wrote the resulting \( n \) products on the board instead of the original numbers. Afterward, Anton wrote down the difference between the largest and the smallest of the numbers Oleksii wrote. Oleksii wants Anton to write the smallest possible number. What is the smallest number that can be written? [i]Proposed by Oleksii Masalitin, Anton Trygub[/i]

An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of the objects in the urn are gold coins? $ \textbf{(A)}\ 40\%\qquad\textbf{(B)}\ 48\%\qquad\textbf{(C)}\ 52\%\qquad\textbf{(D)}\ 60\%\qquad\textbf{(E)}\ 80\% $

A rectangular field has dimensions $120$ meters and $192$ meters. You want to divide it into equal square plots. The measure of the sides of these squares must be an integer number . In addition, you want to place a post in each corner of plot. Determine the smallest number of plots in which you can divide the land and the number of posts needed. [hide=Original wording]Un terreno de forma rectangular de 120 metros por 192 metros se quiere dividir en parcelas cuadradas iguales sin que sobre terreno. La medida de los lados de estos cuadrados debe ser un nu´mero entero. Adem´as se desea colocar un poste en cada esquina de parcela. Determinar el menor nu´mero de parcelas en que se puede dividir el terreno y el nu´mero de postes que se necesitan.[/hide]

For real numbers $ a$ and $ b$, define $ a\$b\equal{}(a\minus{}b)^2$. What is $ (x\minus{}y)^2\$(y\minus{}x)^2$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ x^2\plus{}y^2 \qquad \textbf{(C)}\ 2x^2 \qquad \textbf{(D)}\ 2y^2 \qquad \textbf{(E)}\ 4xy$

A positive integer $k\geqslant 3$ is called[i] fibby[/i] if there exists a positive integer $n$ and positive integers $d_1 < d_2 < \ldots < d_k$ with the following properties: \\ $\bullet$ $d_{j+2}=d_{j+1}+d_j$ for every $j$ satisfying $1\leqslant j \leqslant k-2$, \\ $\bullet$ $d_1, d_2, \ldots, d_k$ are divisors of $n$, \\ $\bullet$ any other divisor of $n$ is either less than $d_1$ or greater than $d_k$. Find all fibby numbers. \\ \\ [i]Proposed by Ivan Novak.[/i]