Permutation $\pi$ of $\{1,\dots,n\}$ is called [b]stable[/b] if the set $\{\pi (k)-k|k=1,\dots,n\}$ is consisted of exactly two different elements. Prove that the number of stable permutation of $\{1,\dots,n\}$ equals to $\sigma (n)-\tau (n)$ in which $\sigma (n)$ is the sum of positive divisors of $n$ and $\tau (n)$ is the number of positive divisors of $n$. Time allowed for this problem was 75 minutes.

Let $ABC$ and $AMN$ be two similar, non-overlapping triangles with the same orientation, such that $AB=AC$ and $AM=AN$. Let $O$ be the circumcentre of the triangle $MAB$. Prove that the points $O, C, N$ and $A$ lie on a circle if and only if the triangle $ABC$ is equilateral.

A two-player game is played on a chessboard with $ m$ columns and $ n$ rows. Each player has only one piece. At the beginning of the game, the piece of the first player is on the upper left corner, and the piece of the second player is on the lower right corner. If two squares have a common edge, we call them adjacent squares. The player having the turn moves his piece to one of the adjacent squares. The player wins if the opponent's piece is on that square, or if he manages to move his piece to the opponent's initial row. If the first move is made by the first player, for which of the below pairs of $ \left(m,n\right)$ there is a strategy that guarantees the second player win. $\textbf{(A)}\ (1998, 1997) \qquad\textbf{(B)}\ (1998, 1998) \qquad\textbf{(C)}\ (997, 1998) \qquad\textbf{(D)}\ (998, 1998) \qquad\textbf{(E)}\ \text{None }$

Find all positive integers $n$ for which it is possible to partition a regular $n$-gon into triangles with diagonals not intersecting inside the $n$-gon such that at every vertex of the $n$-gon an odd number of triangles meet.

Jimmy invites Kima, Lester, Marlo, Namond, and Omar to dinner. There are nine chairs at Jimmy's round dinner table. Jimmy sits in the chair nearest the kitchen. How many different ways can Jimmy's five dinner guests arrange themselves in the remaining $8$ chairs at the table if Kima and Marlo refuse to be seated in adjacent chairs?

[b]3.[/b] A triangulation of a convex closed polygon is the division into triangles of this poilygon by diagonals not intersecting in the interior of the polygon. Find the number of all triangulations fo a conves n-gon and also the number of those triangulations in which every triangle has at least one side in common with the given n-gon. [b](C. 4)[/b]

Let $ABC$ be a triangle inscribed in the circle $(O)$, with orthocenter $H$. Let d be an arbitrary line which passes through $H$ and intersects $(O)$ at $P$ and $Q$. Draw diameter $AA'$ of circle $(O)$. Lines $A'P$ and $A'Q$ meet $BC$ at $K$ and $L$, respectively. Prove that $O, K, L$ and $A'$ are concyclic.

Let $M$ be inside segment $BC$ in triangle $\triangle ABC$. $(ABM)$ cuts $AC$ in $A$ and $N$. Construct the circle through $A,N$ and tangent to $BC$ in $P$. Prove that $\measuredangle BAP=\measuredangle PNM$.

For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$. [*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list] [i]Proposed by Bruno Le Floch, France[/i]

$A$ is the set of points of a convex $n$-gon on a plane. The distinct pairwise distances between any $2$ points in $A$ arranged in descending order is $d_1>d_2>...>d_m>0$. Let the number of unordered pairs of points in $A$ such that their distance is $d_i$ be exactly $\mu _i$, for $i=1, 2,..., m$. Prove: For any positive integer $k\leq m$, $\mu _1+\mu _2+...+\mu _k\leq (3k-1)n$.

Find the number of integers $a$ with $1\le a\le 2012$ for which there exist nonnegative integers $x,y,z$ satisfying the equation \[x^2(x^2+2z) - y^2(y^2+2z)=a.\] [i]Ray Li.[/i] [hide="Clarifications"][list=1][*]$x,y,z$ are not necessarily distinct.[/list][/hide]

(a) Positive rational numbers $a, b,$ and $c$ have the property that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ is an integer. Is it possible for $\frac{a}{c} + \frac{c}{b} + \frac{b}{a}$ to also be an integer except for the trivial solution? (b) Positive real numbers $a, b,$ and $c$ have the property that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ is an integer. Is it possible for $\frac{a}{c} + \frac{c}{b} + \frac{b}{a}$ to also be an integer except for the trivial solution? [i](Andrew Brahms, USA)[/i]

Two straight lines $p, q$ are given in the plane and on the straight line $q$ there is a point $F$, $F \not\in p$. Determine the set of all points $X$ that can be obtained by this construction: In the plane we choose a point $S$ that lies neither on $p$ nor on $q$, and we construct a circle $k$ with center $S$ that is tangent to the line $p$. On the circle $k$ we choose a point $T$ such that so that $ST \parallel q$. If the line $FT$ intersects the line $p$ at the point $U$, $X$ is the intersection of the lines $SU$ and $q$

Does there exist an infinite set $\displaystyle{M}$ consisting of positive integers such that for any $\displaystyle{a,b \in M}$ with $\displaystyle{a < b}$ the sum $\displaystyle{a + b}$ is square-free? [b]Note.[/b] A positive integer is called square-free if no perfect square greater than $\displaystyle{1}$ divides it. [i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]

Circles $O_1, O_2$ intersects at $A, B$. The circumcircle of $O_1BO_2$ intersects $O_1, O_2$ and line $AB$ at $R, S, T$ respectively. Prove that $TR = TS$

$ \odot O_1$ and $ \odot O_2$ meet at points $ P$ and $ Q$. The circle through $ P$, $ O_1$ and $ O_2$ meets $ \odot O_1$ and $ \odot O_2$ at points $ A$ and $ B$. Prove that the distance from $ Q$ to the lines $ PA$, $ PB$ and $ AB$ are equal. (Prove the following three cases: $ O_1$ and $ O_2$ are in the common space of $ \odot O_1$ and $ \odot O_2$; $ O_1$ and $ O_2$ are out of the common space of $ \odot O_1$ and $ \odot O_2$; $ O_1$ is in the common space of $ \odot O_1$ and $ \odot O_2$, $ O_2$ is out of the common space of $ \odot O_1$ and $ \odot O_2$.

$p(x)\in \mathbb{C}[x]$ is a polynomial such that: $\forall z\in \mathbb{C}, |z|=1\Longrightarrow p(z)\in \mathbb{R}$ Prove that $p(x)$ is constant.

$2024$ ones and $2024$ twos are arranged in a circle in some order. Is it always possible to divide the circle into [b]a)[/b] two (contiguous) parts with equal sums? [b]b)[/b] three (contiguous) parts with equal sums? [i]Proposed by Fedir Yudin[/i]

Find constants $A > B$ such that $\frac{f\left( \frac{1}{1+2x}\right) }{f(x)}$ is independent of $x$, where $f(x) = \frac{1 + Ax}{1 + Bx}$ for all real $x \ne - \frac{1}{B}$. Put $a_0 = 1$, $a_{n+1} = \frac{1}{1 + 2a_n}$. Find an expression for an by considering $f(a_0), f(a_1), ...$.

The sequence $(a_n)$ is such that $a_{n+1} = (a_n)^n + n + 1$ for all positive integers $n$, where $a_1$ is some positive integer. Let $k$ be the greatest power of $3$ by which $a_{101}$ is divisible. Find all possible values of $k$. [i]Proposed by Kyrylo Holodnov[/i]

Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $ 1$ Joule of energy to hop one step north or one step south, and $ 1$ Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with $ 100$ Joules of energy, and hops till he falls asleep with $ 0$ energy. How many different places could he have gone to sleep?

Minnie rides on a flat road at $20$ kilometers per hour (kph), downhill at $30$ kph, and uphill at $5$ kph. Penny rides on a flat road at $30$ kph, downhill at $40$ kph, and uphill at $10$ kph. Minnie goes from town $A$ to town $B$, a distance of $10$ km all uphill, then from town $B$ to town $C$, a distance of $15$ km all downhill, and then back to town $A$, a distance of $20$ km on the flat. Penny goes the other way around using the same route. How many more minutes does it take Minnie to complete the $45$-km ride than it takes Penny? $\textbf{(A)}\ 45\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 65\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 95$

Prove that \[ \sqrt{(1^k+2^k)(1^k+2^k+3^k)\ldots (1^k+2^k+\ldots +n^k)}\] \[ \ge 1^k+2^k+\ldots +n^k-\frac{2^{k-1}+2\cdot 3^{k-1}+\ldots + (n-1)\cdot n^{k-1}}{n}\] for all integers $n,k \ge 2$.

Let $ a,b,c $ be positive real numbers such that $ a^4+b^4+c^4=3 $. Prove that \[ \frac{9}{a^2+b^4+c^6}+\frac{9}{a^4+b^6+c^2}+\frac{9}{a^6+b^2+c^4}\leq\ a^6+b^6+c^6+6 \]

A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for $3200$ dollars. But during the first week the quarry only sells $60\%$ of the gravel at full price. The following week the quarry drops the price by $10\%$, and, again, it sells $60\%$ of the remaining gravel. Each week, thereafter, the quarry reduces the price by another $10\%$ and sells $60\%$ of the remaining gravel. This continues until there is only a handful of gravel left. How many dollars does the quarry collect for the sale of all its gravel?