Triangle $JHU$ has side lengths $JH=13$, $HU=14$, and $JU=15$. Point $X$ lies on $\overline{HU}$ such that $\triangle{JHX}$ and $\triangle{JUX}$ have equal perimeters. Compute $JX^2$.

Pixar Prison, for Pixar villains, is shaped like a 600 foot by 1000 foot rectangle with a 300 foot by 500 foot rectangle removed from it, as shown below. The warden separates the prison into three congruent polygonal sections for villains from The Incredibles, Finding Nemo, and Cars. What is the perimeter of each of these sections? [asy] draw((0,0)--(0,6)--(10,6)--(10,0)--(8,0)--(8,3)--(3,3)--(3,0)--(0,0)); label("600", (1,3.5)); label("1000", (5.5,6.5)); label("300", (4,1.5)); label("500", (5.5,3.5)); label("300", (1.5,-0.5)); [/asy] [i]Proposed by Peter Rowley

Alberto chooses $2022$ integers $a_1,\,a_2,\dots,\,a_{2022}$ (not necessarily positive and not necessarily distinct) and places them on a $2022\times 2022$ table such that in the $(i,j)$ cell is the number $a_k$, with $k=\max\{i,j\}$, as shown in figure (in which, for a better readability, we have denoted $a_{2022}$ with $a_n$). Barbara does not know the numbers Alberto has chosen, but knows how they are displaced in the table. Given a positive integer $k$, with $1\leq k\leq 2022$, Barbara wants to determine the value of $a_k$ (and she is not interested in determining the values of the other $a_i$'s with $i\neq k$). To do so, Barbara is allowed to ask Alberto one or more questions, in each of which she demands the value of the sum of the numbers contained in the cells of a "path", where with the term "path" we indicate a sorted list of cells with the following characteristics: • the path starts from the top left cell and finishes with the bottom right cell, • the cells of the path are all distinct, • two consecutive cells of the path share a common side. Determine, as $k$ varies, the minimum number of questions Barbara needs to find $a_k$.

A complete set of the Encyclopedia of Mathematics has $10$ volumes. There are ten mathematicians in Mathemagic Land, and each of them owns two volumes of the Encyclopedia. Together they own two complete sets. Show that there is a way for each mathematician to donate one book to the library such that the library receives a complete set.

Let $a_1,a_2,\dots$ be a sequence of real numbers satisfying $$\frac{a_{n+1}}{a_n}-\frac{a_{n+2}}{a_n}-\frac{a_{n+1}a_{n+2}}{a_n^2}=\frac{na_{n+2}a_{n+1}}{a_n}.$$ Given that $a_1=-1$ and $a_2=-\tfrac{1}{2}$, find the value of $\tfrac{a_9}{a_{20}}$.

Given a polynomial $P(x)$ with positive real coefficients. Prove that $P(1)P(xy) \ge P(x)P(y)$ for all $x\ge1, y \ge 1$. (K. Gorodnin)

The sequences of natural numbers $p_n$ and $q_n$ are given such that $$p_1 = 1,\ q_1 = 1,\ p_{n + 1} = 2q_n^2-p_n^2,\ q_{n + 1} = 2q_n^2+p_n^2 $$ Prove that $p_n$ and $q_m$ are coprime for any m and n.

In a tournament with $n$ players, $n$ < 10, each player plays once against each other player scoring 1 point for a win and 0 points for a loss. Draws do not occur. In a particular tournament only one player ended with an odd number of points and was ranked fourth. Determine whether or not this is possible. If so, how many wins did the player have?

Let $p$ be an odd prime number. Find all natural numbers $k$ such that $$\sqrt{k^2 - pk}$$ is a positive integer.

Let $d(n)$ denote the number of positive divisors of $n$ and let $e(n)=\left[2000\over n\right]$ for positive integer $n$. Prove that \[d(1)+d(2)+\dots+d(2000)=e(1)+e(2)+\dots+e(2000).\]

Prove the inequality \[ {x\over y^2-z}+{y\over z^2-x}+{z\over x^2-y} > 1, \] where $2 < x, y, z < 4.$ [i]Proposed by A. Golovanov[/i]

Patrick and Anderson are having a snowball fight. Patrick throws a snowball at Anderson which is shaped like a sphere with a radius of $10$ centimeters. Anderson catches the snowball and uses the snow from the snowball to construct snowballs with radii of $4$ centimeters. Given that the total volume of the snowballs that Anderson constructs cannot exceed the volume of the snowball that Patrick threw, how many snowballs can Anderson construct?

Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$; (ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$. Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list] [i]Proposed by Hans Zantema, Netherlands[/i]

Let $P$ be a point inside or outside (but not on) of a triangle $ABC$. Prove that $PA +PB +PC$ is greater than half of the perimeter of the triangle

Let $\mathcal{S}$ be the set of points $(x, y)$ in the Cartesian coordinate plane such that $xy > 0$ and $x^2 + y^2 + 2x + 4y \leq 2021.$ The total area of $\mathcal{S}$ can be written in the form $a\pi + b,$ where $a$ and $b$ are integers. Compute $a + b.$

Given a circle and a point $ C$ not lying on this circle. Consider all triangles $ ABC$ such that points $ A$ and $ B$ lie on the given circle. Prove that the triangle of maximal area is isosceles.

Let $a$ and $ b$ be positive integers bigger than $2$. Prove that there exists a positive integer $k$ and a sequence $n_1, n_2, ..., n_k$ consisting of positive integers, such that $n_1 = a,n_k = b$, and $(n_i + n_{i+1}) | n_in_{i+1}$ for all $i = 1,2,..., k - 1$

What is the maximum number of balls of clay of radius $2$ that can completely fit inside a cube of side length $6$ assuming the balls can be reshaped but not compressed before they are packed in the cube? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$

Given a sequence $\{a_n\}$: $a_1=1, a_{n+1}=\left\{ \begin{array}{lcr} a_n+n,\quad a_n\le n, \\ a_n-n,\quad a_n>n, \end{array} \right. \quad n=1,2,\cdots.$ Find the number of positive integers $r$ satisfying $a_r<r\le 3^{2017}$.

Let $\Delta ABC$ be a triangle with an inscribed circle centered at $I$. The line perpendicular to $AI$ at $I$ intersects $\odot (ABC)$ at $P,Q$ such that, $P$ lies closer to $B$ than $C$. Let $\odot (BIP) \cap \odot (CIQ) =S$. Prove that, $SI$ is the angle bisector of $\angle PSQ$

Let $ x_1 , x_2, x_3 ...$ a succession of positive integers such that for every couple of positive integers $(m,n)$ we have $ x_{mn} \neq x_{m(n+1)}$ . Prove that there exists a positive integer $i$ such that $x_i \ge 2017 $.

Given a square $ABCD$ with side length $6$. We draw line segments from the midpoints of each side to the vertices on the opposite side. For example, we draw line segments from the midpoint of side $AB$ to vertices $C$ and $D$. The eight resulting line segments together bound an octagon inside the square. What is the area of this octagon?

For which real numbers $p$ does the equation $x^{2}+px+3p=0$ have integer solutions ?

Given integer $a_1\geq 2$. For integer $n\geq 2$, define $a_n$ to be the smallest positive integer which is not coprime to $a_{n-1}$ and not equal to $a_1,a_2,\cdots, a_{n-1}$. Prove that every positive integer except 1 appears in this sequence $\{a_n\}$.

The factors of $ x^4\plus{}64$ are: $ \textbf{(A)}\ (x^2\plus{}8)^2 \qquad \textbf{(B)}\ (x^2\plus{}8)(x^2\minus{}8) \qquad \textbf{(C)}\ (x^2\plus{}2x\plus{}4)(x^2\minus{}8x\plus{}16) \\ \textbf{(D)}\ (x^2\minus{}4x\plus{}8)(x^2\minus{}4x\minus{}8) \qquad \textbf{(E)}\ (x^2\minus{}4x\plus{}8)(x^2\plus{}4x\plus{}8)$