There are $9$ vertical columns in a row. In some places, horizontal sticks are inserted between adjacent columns, no two are at the same height. The beetle crawls from the bottom up; when he meets the wand, he crawls over it to the next column and continues to crawl up. It is known that if a beetle starts at the bottom of the first (leftmost) column, then it will end its journey on the ninth (rightmost) column. Is it always possible to remove one stick so that the beetle ends up at the top of the fifth column? (For example, if the sticks are arranged as in picture, the beetle will crawl along a solid line. If you remove the third one A stick in the path of the beetle, then it will crawl along the dotted line.) [i] Proposed by G. Karavaev[/i]

In a right-angled triangle, the sum $a + b$ of the sides enclosing the right angle equals $24$ while the length of the altitude $h_c$ on the hypotenuse $c$ is $7$. Determine the length of the hypotenuse.

A circle is inscribed in triangle $ABC$. $M$ and $N$ are the points of its tangency with the sides $BC$ and $CA$, respectively. The segment $AM$ intersects $BN$ at point $P$ and the inscribed circle at point $Q$. It is known that $MP = a$, $PQ = b$. Find $AQ$.

Let $a_1,a_2,\ldots, a_{2013}$ be a permutation of the numbers from $1$ to $2013$. Let $A_n = \frac{a_1 + a_2 + \cdots + a_n} {n}$ for $n = 1,2,\ldots, 2013$. If the smallest possible difference between the largest and smallest values of $A_1,A_2,\ldots, A_{2013}$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. [i]Ray Li[/i]

Two sequences $(a_n)$ and $(b_n)$ satisfy that $a_0=1,a_1=4,a_2=49$, and $\begin{cases} a_{n+1}=7a_n+6b_n-3\\ b_{n+1}=8a_n+7b_n-4\\ \end{cases}$ for $n=0,1,2,\cdots,$. Prove that $a_n$ is a perfect square for $n=0,1,2,\cdots,$.

When we cut a rope into two pieces, we say that the cut is special if both pieces have different lengths. We cut a chord of length $2008$ into two pieces with integer lengths and we write those lengths on the board. Afterwards, we cut one of the pieces into two new pieces with integer lengths and we write those lengths on the board. This process ends until all pieces have length $1$. $a)$ Find the minimum possible number of special cuts. $b)$ Prove that, for all processes that have the minimum possible number of special cuts, the number of different integers on the board is always the same.

Let $A,B$ be $n\times n$ matrices, $n\geq 2$, and $B^2=B$. Prove that: $$\text{rank}\,(AB-BA)\leq \text{rank}\,(AB+BA)$$

(a)The game of "super- chess" is played on a $30 \times 30$ board and involves $20$ different pieces. Each piece moves according to its own rules , but cannot move from any square to more than $20$ other squares . A piece "captures" another piece which is on a square to which it has moved. A permitted move (e.g. $m$ squares forward and $n$ squares to the right) does not depend on the piece 's starting square . Prove that (i) A piece cannot cap ture a piece on a given square from more than $20$ starting squares. (ii) It is possible to arrange all $20$ pieces on the board in such a way that not one of them can capture any of the others in one move. (b) The game of "super-chess" is played on a $100 \times 100$ board and involves $20$ different pieces. Each piece moves according to its own rules , but cannot move from any square to more than $20$ other squares. A piece "captures" another piece which is on a square to which it has moved. It is possible that a permitted move (e.g. $m$ squares forward and $n$ squares to the right) may vary, depending on the piece's position . Prove that one can arrange all $20$ pieces on the board in such a way that not one of them can capture any of the others in one move. ( A . K . Tolpygo, Kiev) PS. (a) for Juniors , (b) for Seniors

Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that: (i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$. (ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$. [i]Proposed by Yang Liu[/i]

For a scalene triangle $ABC$ with an incenter $I$, let its incircle meets the sides $BC, CA, AB$ at $D, E, F$, respectively. Denote by $P$ the intersection of the lines $AI$ and $DF$, and $Q$ the intersection of the lines $BI$ and $EF$. Prove that $\overline{PQ}=\overline{CD}$.

Determine all pairs $(p,q)$ of primes for which both $p^2+q^3$ and $q^2+p^3$ are perfect squares.

As shown in figure , in the pentagon $ABCDE$, $BC = DE$, $CD \parallel BE$, $AB>AE$. If $\angle BAC = \angle DAE$ and $\frac{AB}{BD}=\frac{AE}{ED}$. Prove that $AC$ bisects the line segment $BE$. [img]https://cdn.artofproblemsolving.com/attachments/3/2/5ce44f1e091786b865ae4319bda56c3ddbb8d7.png[/img]

Find all tuples $ (x_1,x_2,x_3,x_4,x_5,x_6)$ of non-negative integers, such that the following system of equations holds: $ x_1x_2(1\minus{}x_3)\equal{}x_4x_5 \\ x_2x_3(1\minus{}x_4)\equal{}x_5x_6 \\ x_3x_4(1\minus{}x_5)\equal{}x_6x_1 \\ x_4x_5(1\minus{}x_6)\equal{}x_1x_2 \\ x_5x_6(1\minus{}x_1)\equal{}x_2x_3 \\ x_6x_1(1\minus{}x_2)\equal{}x_3x_4$

Let $n$ be an integer greater than $1$ such that $n$ could be represented as a sum of the cubes of two rational numbers, prove that $n$ is also the sum of the cubes of two non-negative rational numbers. Proposed by [i]Navid Safaei[/i]

This is rather simple, but I liked it :). Show that a disk cannot be partitioned into two congruent subsets.

There are given 2050 points in a unit cube. Show that there are 5 points lying in an (open) ball with the radius 1/9.

Let $f$ be a real function such that for all $x\ne 0$, $x\ne 1$, $$f (x) + f \left(- \frac{1}{x - 1} \right) =\frac{9}{4x^2} + f\left(1 - \frac{1}{x} \right) .$$ Compute $f \left( \frac{1}{2}\right).$ .

Let be given an acute triangle $ABC$. Show that there exist unique points $A_1 \in BC$, $B_1 \in CA$, $C_1 \in AB$ such that each of these three points is the midpoint of the segment whose endpoints are the orthogonal projections of the other two points on the corresponding side. Prove that the triangle $A_1B_1C_1$ is similar to the triangle whose side lengths are the medians of $\triangle ABC$.

Bertha has $ 6$ daughters and no sons. Some of her daughters have $ 6$ daughters, and the rest have none. Bertha has a total of $ 30$ daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no children? $ \textbf{(A)}\ 22 \qquad \textbf{(B)}\ 23 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 25\qquad \textbf{(E)}\ 26$

Let $G_0, G_1,\ldots$ be infinite open subsets of a Hausdorff space. Prove that there exist some infinite pairwise disjoint open sets $V_0,V_1,\ldots$ and some indices $n_0<n_1<\cdots$ such that $V_i\subseteq G_{n_i}$ for every $i\geqslant 0.$

I have a four-digit palindrome $\underline{a} \ \underline{b} \ \underline{b} \ \underline{a}$ that is divisible by $b$ and is also divisible by the two-digit number $\underline{b} \ \underline{b}.$ Find the number of palindromes satisfying both of these properties.

For a natural number $ n\ge 2, $ calculate the integer part of $ \sqrt[n]{1+n}-\sqrt {2/n} . $ [i]Dan Nedeianu[/i]

How many multiples of $12$ divide $12!$ and have exactly $12$ divisors? [i]Proposed by Adam Bertelli[/i]

Robert tiles a $420 \times 420$ square grid completely with $1 \times 2$ blocks, then notices that the two diagonals of the grid pass through a total of $n$ blocks. Find the sum of all possible values of $n$.

Let us say that a pair of distinct positive integers is nice if their arithmetic mean and their geometric mean are both integer. Is it true that for each nice pair there is another nice pair with the same arithmetic mean? (The pairs $(a, b)$ and $(b, a)$ are considered to be the same pair.) [i]Boris Frenkin[/i]