If $x,y,z$ are real numbers such that $xyz=1$, evaluate $$\frac{x+1}{xy+x+1}+\frac{y+1}{yz+y+1}+\frac{z+1}{zx+z+1}.$$

We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ . [i]Proposed by Abbas Mehrabian, Iran[/i]

Given a convex polyhedron and a sphere intersecting each its edge at two points so that each edge is trisected (divided into three equal parts). Is it necessarily true that all faces of the polyhedron are (a) congruent polygons? (b) regular polygons?

Let $r,s,t$ be the roots of $x^3+6x^2+7x+8$. Find $$(r^2+s+t)(s^2+t+r)(t^2+r+s).$$ [i]Proposed by Evan Chang (squareman), USA[/i]

Peter has many squares of equal side. Some of the squares are black, some are white. Peter wants to assemble a big square, with side equal to $n$ sides of the small squares, so that the big square has no rectangle formed by the small squares such that all the squares in the vertices of the rectangle are of equal colour. How big a square is Peter able to assemble?

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ is a square and $AA'\parallel BB'\parallel CC'\parallel DD'$). Furthermore, let $\mathcal R$ be a rotation (with respect some line) that maps vertex $A$ to $B.$ Find the set of all images $X=\mathcal R(C)$ such that $X$ lies on the surface of the cube for some rotation $\mathcal R(A)=B.$

Let $ a, b \in \mathbb{N}$ with $ 1 \leq a \leq b,$ and $ M \equal{} \left[\frac {a \plus{} b}{2} \right].$ Define a function $ f: \mathbb{Z} \mapsto \mathbb{Z}$ by \[ f(n) \equal{} \begin{cases} n \plus{} a, & \text{if } n \leq M, \\ n \minus{} b, & \text{if } n >M. \end{cases} \] Let $ f^1(n) \equal{} f(n),$ $ f_{i \plus{} 1}(n) \equal{} f(f^i(n)),$ $ i \equal{} 1, 2, \ldots$ Find the smallest natural number $ k$ such that $ f^k(0) \equal{} 0.$

Let $ [a]$ be the integer such that $ [a]\le a<[a]\plus{}1$. Find all real numbers $ (a,b,c)$ such that \[ \{a\}\plus{}[b]\plus{}\{c\}\equal{}2.9\\\{b\}\plus{}[c]\plus{}\{a\}\equal{}5.3\\\{c\}\plus{}[a]\plus{}\{b\}\equal{}4.0.\]

A positive integer $n$ is [i]stacked[/i] if $2n$ has the same number of digits as $n$ and the digits of $2n$ are multiples of the corresponding digits of $n.$ For example, $1203$ is stacked because $2 \times 1203 = 2406,$ and $2, 4, 0, 6$ are multiples of $1, 2, 0, 3,$ respectively. Compute the number of stacked integers less than $1000.$

In how many ways a cube can be painted using seven different colors in such a way that no two faces are in same color? $ \textbf{(A)}\ 154 \qquad\textbf{(B)}\ 203 \qquad\textbf{(C)}\ 210 \qquad\textbf{(D)}\ 240 \qquad\textbf{(E)}\ \text{None of the above} $

Find all positive integers $n$ such that $4^n+6^n+9^n$ is a square. [i]David Yang, Alex Zhu.[/i]

Let $S$ be a set of integers which has the following properties: 1) There exists $x,y\in S$ such that $(x,y)=(x-2,y-2)=1$; 2) For $\forall$ $x,y\in S, x^2-y\in S$. Prove that $S\equiv \mathbb{Z}$ .

$ABCDE$ is a regular pentagon (with vertices in that order) inscribed in a circle of radius $1$. Find $AB \cdot AC$.

Define $f(x) = x^2 + 5$. Find the product of all $x$ such that $f(x) = 14$. $\textbf{(A) }{-}9\qquad\textbf{(B) }{-}3\qquad\textbf{(C) }0\qquad\textbf{(D) }3\qquad\textbf{(E) }9$

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 \[d(1)+d(2)+\dots+d(2000)=e(1)+e(2)+\dots+e(2000).\]