Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 1}$ be sets having $ n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $ n$ can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly $ \frac {n}{2}$ zeros?

Starting from an equilateral triangle with perimeter $P_0$, we carry out the following iterations: the first iteration consists of dividing each side of the triangle into three segments of equal length, construct an exterior equilateral triangle on each of the middle segments, and then remove these segments (bases of each new equilateral triangle formed). The second iteration consists of apply the same process of the first iteration on each segment of the resulting figure after the first iteration. Successively, follow the other iterations. Let $A_n$ be the area of the figure after the $n$- th iteration, and let $P_n$ the perimeter of the same figure. If $A_n = P_n$, find the value of $P_0$ (in its simplest form).

On a circle, 2022 points are chosen such that distance between two adjacent points is always the same. There are $k$ arcs, each having endpoints on chosen points, with different lengths. Arcs do not contain each other. What is the maximum possible number of $k$?

Initially, a positive integer $N$ is written on a blackboard. We repeatedly replace the number according to the following rules: 1) replace the number by a positive multiple of itself 2) replace the number by a number with the same digits in a different order. (The new number is allowed to have leading digits, which are then deleted.) [i]A possible sequence of moves is given by $5 \to 20 \to 140 \to 041=41$.[/i] Determine for which values of $N$ it is possible to obtain $1$ after a finite number of such moves.

The permutations of $OLYMPIAD$ are arranged in lexicographical order, with $ADILMOPY$ being arrangement 1 and its reverse being arrangement $40320$. Yu Semo and Yu Sejmo both choose a uniformly random arrangement. The immature Yu Sejmo exclaims, ``My fourth letter is $L$!" while Yu Semo remains silent. Given this information, let $E_1$ be the expected arrangement number of Yu Semo and $E_2$ be the expected arrangement number of Yu Sejmo. Compute $E_2 - E_1$.

In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation \[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \] Prove that $a_{1}=a_{2}= \ldots= a_{n}$.

A game board consists of $64$ squares that alternate in color between black and white. The figure below shows square $P$ in the bottom and square $Q$ in the top row. A marker is placed at $P$. A [i]step[/i] consists of moving the marker onto one of the adjoining white squares in the row above. How many $7$-step paths are there from $P$ to $Q$? (The figure shows a sample path.) [asy]//diagram by SirCalcsALot size(200); int[] x = {6, 5, 4, 5, 6, 5, 6}; int[] y = {1, 2, 3, 4, 5, 6, 7}; int N = 7; for (int i = 0; i < 8; ++i) { for (int j = 0; j < 8; ++j) { draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)); if ((i+j) % 2 == 0) { filldraw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--(i,j)--cycle,black); } } } for (int i = 0; i < N; ++i) { draw(circle((x[i],y[i])+(0.5,0.5),0.35)); } label("$P$", (5.5, 0.5)); label("$Q$", (6.5, 7.5)); [/asy] $\textbf{(A)}\ 28 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 35$

Let a non-isosceles acute triangle ABC with tha attitude AD, BE, CF and the orthocenter H. DE, DF intersect (AD) at M, N respectively. $P\in AB,Q\in AC$ satisfy $NP\perp AB,MQ\perp AC$ a) Prove that EF is the tangent line of (APQ) b) Let T be the tangency point of (APQ) with EF,.DT $\cap$ MN={K}. L is the reflection of A in MN. Prove that MN, EF ,(DLK) pass through a piont

We define the following sequence: $a_0=a_1=1$, $a_{n+1}=14a_n-a_{n-1}$. Prove that $2a_n-1$ is a perfect square.

Let $ABCD$ be an inscribed quadrilateral whose diagonals are connected internally. are perpendicular to each other and intersect at the point $P$. Prove that the line connecting the midpoints of the opposite sides of the quadrilateral $ABCD$ bisects the lines $OP$ ($O$ is the center of the circle circumscribed around quadrilateral $ABCD$). (Alexander Dunyak)

An integer $x$ is selected at random between 1 and $2011!$ inclusive. The probability that $x^x - 1$ is divisible by $2011$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$. [i]Author: Alex Zhu[/i]

Let $AB$ and $AC$ be two distinct rays not lying on the same line, and let $\omega$ be a circle with center $O$ that is tangent to ray $AC$ at $E$ and ray $AB$ at $F$. Let $R$ be a point on segment $EF$. The line through $O$ parallel to $EF$ intersects line $AB$ at $P$. Let $N$ be the intersection of lines $PR$ and $AC$, and let $M$ be the intersection of line $AB$ and the line through $R$ parallel to $AC$. Prove that line $MN$ is tangent to $\omega$. [i]Warut Suksompong, Thailand[/i]

Determine all functions $f : R\to R$ satisfying: $$f(xy + 2x + 2y - 1) = f(x)f(y) + f(y) + x -2$$ for all real numbers $x, y$.

Find the tangents of the angles of a triangle knowing that they are positive integers.

Is there a strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x?$

Two congruent rectangles are located as shown in the figure. Find the area of the shaded part. [img]https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png[/img]

Prove that any triangle can be cut into $2019$ quadrilaterals such that each quadrilateral is both inscribed and circumscribed. (Nairi Sedrakyan)

There are $172$ two-way direct airways between $20$ cities, at most one between any two cities. Prove that one can reach any city from any other city with at most one transfer.

Let $M = \{1,2,\cdots , 10\}$, and let $T$ be a set of 2-element subsets of $M$. For any two different elements $\{a,b\}, \{x,y\}$ in $T$, the integer $(ax+by)(ay+bx)$ is not divisible by 11. Find the maximum size of $T$.

Compute the number of ordered pairs $(m,n)$ of positive integers that satisfy the equation $\text{lcm}(m,n)+\gcd(m,n)=m+n+30$. [i]Proposed by Ankit Bisain[/i]

Let $p$ be a prime and suppose $2^{2p} \equiv 1 (\text{mod}$ $ 2p+1)$ is prime. Prove that $2p+1$ is prime$^{1}$ [size=75]$^{1}$This is a special case of Pocklington's theorem. A proof of this special case is required.[/size]

Do there exist infinitely many triples of positive integers $(a, b, c)$ such that $a$, $b$, $c$ are pairwise coprime, and $a! + b! + c!$ is divisible by $a^2 + b^2 + c^2$? [i]Proposed by Anzo Teh Zhao Yang[/i]

Quadrilateral $ABCD$ is both cyclic and circumscribed. Its incircle touches its sides $AB$ and $CD$ at points $X$ and $Y$, respectively. The perpendiculars to $AB$ and $CD$ drawn at $A$ and $D$, respectively, meet at point $U$; those drawn at $X$ and $Y$ meet at point $V$, and finally, those drawn at $B$ and $C$ meet at point $W$. Prove that points $U$, $V$ and $W$ are collinear. [i]Proposed by A. Golovanov[/i]

Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end? ${ \textbf{(A)}\ 62 \qquad\textbf{(B)}\ 82 \qquad\textbf{(C)}\ 83\qquad\textbf{(D}}\ 102\qquad\textbf{(E)}\ 103 $

The two intersections of the circles $k_i$ and $k_{i + 1}$ are $P_i$ and $Q_i$ ($1 \le i \le 5, k_6 = k_1$). On the circle $k_1$ lies an arbitrary point $A$. Then the points $B, C, D, E, F, G, H, I, J, K$ lie on the circles $k_2, k_3, k_4, k_5, k_1, k_2, k_3, k_4, k_5, k_1$ respectively, such that $AP_1B, BP_2C, CP_3D, DP_4E, EP_5F, F Q_1G, GQ_2H, HQ_3I, IQ_4J, JQ_5K$ are straight line triplets. Prove that that $K = A$. [img]https://1.bp.blogspot.com/-g6rF1hcPE08/X9j1SEJT7-I/AAAAAAAAMzc/2rWIiWTHZ34zfWVeGujkCxRW1hSCw5oOwCLcBGAsYHQ/s16000/2016%2BDurer%2BC..4.png[/img] [i]Circles can have different radii, and They can be located in different ways from the figure. We assume that during editing none neither of the two points mentioned above coincide.[/i]