Define the sequence $ a_1, a_2, a_3, ...$ as follows. $ a_1$ and $ a_2$ are coprime positive integers and $ a_{n \plus{} 2} \equal{} a_{n \plus{} 1}a_n \plus{} 1$. Show that for every $ m > 1$ there is an $ n > m$ such that $ a_m^m$ divides $ a_n^n$. Is it true that $ a_1$ must divide $ a_n^n$ for some $ n > 1$?

Peer and Solveig are playing a game with $n$ coins, all of which show $M$ on one side and $K$ on the opposite side. The coins are laid out in a row on the table. Peer and Solveig alternate taking turns. On his turn, Peer may turn over one or more coins, so long as he does not turn over two adjacent coins. On her turn, Solveig picks precisely two adjacent coins and turns them over. When the game begins, all the coins are showing $M$. Peer plays first, and he wins if all the coins show $K$ simultaneously at any time. Find all $n\geqslant 2$ for which Solveig can keep Peer from winning.

Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$ has radius $r^2$ and is internally tangent to $C_1$ at point $A_1$. In this way a sequence of circles $C_1,C_2,C_3,...$ and a sequence of points on the circles $A_1,A_2,A_3,...$ are constructed, where circle $C_n$ has radius $r^n$ and is internally tangent to circle $C_{n-1}$ at point $A_{n-1}$, and point $A_n$ lies on $C_n$ $90^{\circ}$ counterclockwise from point $A_{n-1}$, as shown in the figure below. There is one point $B$ inside all of these circles. When $r=\frac{11}{60}$, the distance from the center of $C_0$ to $B$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] size(6cm); real r = 0.8; pair nthCircCent(int n){ pair ans = (0, 0); for(int i = 1; i <= n; ++i) ans += rotate(90 * i - 90) * (r^(i - 1) - r^i, 0); return ans; } void dNthCirc(int n){ draw(circle(nthCircCent(n), r^n)); } dNthCirc(0); dNthCirc(1); dNthCirc(2); dNthCirc(3); dot("$A_0$", (1, 0), dir(0)); dot("$A_1$", nthCircCent(1) + (0, r), dir(135)); dot("$A_2$", nthCircCent(2) + (-r^2, 0), dir(0)); [/asy]

Find all positive integers $N$ such that an $N\times N$ board can be tiled using tiles of size $5\times 5$ or $1\times 3$. Note: The tiles must completely cover all the board, with no overlappings.

Find all functions $f : R \to R$ satisfying the conditions: 1. $f (x + 1) \ge f (x) + 1$ for all $x \in R$ 2. $f (x y) \ge f (x)f (y)$ for all $x, y \in R$

$$\int_{0}^{\frac{\pi}{2}}\sin^2(x)\sin(2x)\,dx$$ [i]Proposed by Connor Gordon[/i]

Find all polynomials $P(x)$ with real coefficients such that \[P(a)\in\mathbb{Z}\ \ \ \text{implies that}\ \ \ a\in\mathbb{Z}.\]

One hundred red points and one hundred blue points are chosen in the plane, no three of them lying on a line. Show that these points can be connected pairwise, red ones with blue ones, by disjoint line segments.

[b]interesting function[/b] $S$ is a set with $n$ elements and $P(S)$ is the set of all subsets of $S$ and $f : P(S) \rightarrow \mathbb N$ is a function with these properties: for every subset $A$ of $S$ we have $f(A)=f(S-A)$. for every two subsets of $S$ like $A$ and $B$ we have $max(f(A),f(B))\ge f(A\cup B)$ prove that number of natural numbers like $x$ such that there exists $A\subseteq S$ and $f(A)=x$ is less than $n$. time allowed for this question was 1 hours and 30 minutes.

We say that a quadruple $(A,B,C,D)$ is [i]dobarulho[/i] when $A,B,C$ are non-zero algorisms and $D$ is a positive integer such that: $1.$ $A \leq 8$ $2.$ $D>1$ $3.$ $D$ divides the six numbers $\overline{ABC}$, $\overline{BCA}$, $\overline{CAB}$, $\overline{(A+1)CB}$, $\overline{CB(A+1)}$, $\overline{B(A+1)C}$. Find all such quadruples.

There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed to choose a player whose each neighbor holds a nonzero number of cards. The game terminates when there is no such player. Determine the largest possible value of $K$ such that, no matter how we deal the cards and how we choose the players, the game always terminates after a finite number of turns. [i]Proposed by Peter Novotný, Slovakia[/i]

Can every positive rational number $q$ be written as $$\frac{a^{2021} + b^{2023}}{c^{2022} + d^{2024}},$$ where $a, b, c, d$ are all positive integers? [i]Proposed by Dominic Yeo, UK[/i]

Let be three positive real numbers $ a,b,c $ whose sum is $ 1. $ Prove: $$ 0\le\sum_{\text{cyc}} \log_a\frac{(abc)^a}{a^2+b^2+c^2} $$

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Say that a rational number is [i]special[/i] if its decimal expansion is of the form $0.\overline{abcdef}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are digits (possibly equal) that include each of the digits $2$, $0$, $1$, and $5$ at least once (in some order). How many special rational numbers are there?

There is a unique positive real number $x$ such that the three numbers $\log_8(2x),\log_4x,$ and $\log_2x,$ in that order, form a geometric progression with positive common ratio. The number $x$ can be written as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A deck of $100$ cards is labeled $1,2,\ldots,100$ from top to bottom. The top two cards are drawn; one of them is discarded at random, and the other is inserted back at the bottom of the deck. This process is repeated until only one card remains in the deck. Compute the expected value of the label of the remaining card.

Suppose there are $4n$ line segments of unit length inside a circle of radius $n$. Furthermore, a straight line $L$ is given. Prove that there exists a straight line $L'$ that is either parallel or perpendicular to $L$ and that $L'$ cuts at least two of the given line segments.

A circle of radius $R$ is inscribed into an acute triangle. Three tangents to the circle split the triangle into three right angle triangles and a hexagon that has perimeter $Q$. Find the sum of diameters of circles inscribed into the three right triangles. (6)

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$. Let the lines $AB$ and $CD$ intersect at $P$, and the lines $AD$ and $BC$ intersect at $Q$. Let then the circumcircle of the triangle $\triangle APQ$ intersect $\Omega$ at $R \neq A$. Prove that the line $CR$ goes through the midpoint of the segment $PQ$.

Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area. [img]https://1.bp.blogspot.com/-xt3pe0rp1SI/XzcGLgEw1EI/AAAAAAAAMYM/vFKxvvVuLvAJ5FO_yX315X3Fg_iFaK2fACLcBGAsYHQ/s0/1997%2BMohr%2Bp2.png[/img]

Suppose the integers $1,2,\ldots 10$ are split into two disjoint collections $a_1,a_2, \ldots a_5$ and $b_1 , \ldots b_5$ such that $a_1 <a _2 < a_3 <a_4 <a _5 , b_1 < b_2 < b_3 < b_4 < b_5$ (i) Show that the larger number in any pair $\{ a_j, b_j \}$ , $1 \leq j \leq 5$ is at least $6$. (ii) Show that $\sum_{i=1} ^{5} | a_i - b_i|$ = 25 for every such partition.

Let $n \ge 3$ be an integer, and consider a $n \times n$-board, divided into $n^2$ unit squares. For all $m \ge 1$, arbitrarily many $1\times m$-rectangles (type I) and arbitrarily many $m\times 1$-rectangles (type II) are available. We cover the board with $N$ such rectangles, without overlaps, and such that every rectangle lies entirely inside the board. We require that the number of type I rectangles used is equal to the number of type II rectangles used.(Note that a $1 \times 1$-rectangle has both types.) What is the minimal value of $N$ for which this is possible?

Find all pairs $(m,n)$ of integers that satisfy the equation \[(m-n)^{2}=\frac{4mn}{m+n-1}.\]

Given a prime $p$, find $\gcd(\binom{2^pp}{1},\binom{2^pp}{3},\ldots, \binom{2^pp}{2^pp-1}) $.