Given a acute triangle $PA_1B_1$ is inscribed in the circle $\Gamma$ with radius $1$. for all integers $n \ge 1$ are defined: $C_n$ the foot of the perpendicular from $P$ to $A_nB_n$ $O_n$ is the center of $\odot (PA_nB_n)$ $A_{n+1}$ is the foot of the perpendicular from $C_n$ to $PA_n$ $B_{n+1} \equiv PB_n \cap O_nA_{n+1}$ If $PC_1 =\sqrt{2}$, find the length of $PO_{2015}$ [hide=Source]Cono Sur Olympiad - 2015 - Day 1 - Problem 3[/hide]

A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors). Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.) [hide="Problem 8 from CWMO 2007"]$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.[/hide]

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible. [i]Proposed by Horst Sewerin, Germany[/i]

Determine all positive integers $n\geq 2$ for which there exist integers $x_1,x_2,\ldots ,x_{n-1}$ satisfying the condition that if $0<i<n,0<j<n, i\neq j$ and $n$ divides $2i+j$, then $x_i<x_j$.

Find the number of arithmetic sequences $a_1,a_2,a_3$ of three nonzero integers such that the sum of the terms in the sequence is equal to the product of the terms in the sequence. [i]Proposed by Sammy Charney[/i]

Let $c$ be a positive real number. Prove that $c$ can be expressed in infinitely many ways as a sum of infinitely many distinct terms selected from the sequence $\left( \frac{1}{10n} \right)_{n\in \mathbb{N}}$

If $0 \leq a \leq b \leq c \leq d,$ prove that \[a^bb^cc^dd^a \geq b^ac^bd^ca^d.\]

A museum has the shape of a $n \times n$ square divided into $n^2$ rooms of the shape of a unit square $(n>1)$. Between every two adjacent rooms (i.e. sharing a wall) there is a door. A night guardian wants to organize an inspection journey through the museum according to the following rules. He starts from some room and, whenever he enters a room, he stays there for exactly one minute and then proceeds to another room. He is allowed to enter a room more than once, but at the end of his journey he must have spent exactly $k$ minutes in every room. Find all $n$ and $k$ for which it is possible to organize such a journey.

Function $f(x)=\frac{x}{1-2^x}-\frac{x}{2}$ is $\text{(A)}$ an even function, not an odd function. $\text{(B)}$ an odd function, not an even function. $\text{(C)}$ an even function, also an odd function. $\text{(D)}$ neither an even function, nor an odd function.

Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m -3n)^2 = 2019$. Find $3m + n$.

Let $p$ be a prime number. Call a positive integer $n$ interesting if \[x^n-1=(x^p-x+1)f(x)+pg(x)\] for some polynomials $f$ and $g$ with integer coefficients. a) Prove that the number $p^p-1$ is interesting. b) For which $p$ is $p^p-1$ the minimal interesting number?

In triangle $ABC$ let $N, M, P$ be the midpoints of the sides $BC, CA, AB$ and $G$ be the centroid of this triangle. Let the circle circumscribed to $BGP$ intersect the line $MP$ in point $K$, $P \neq K$, and the circle circumscribed to $CGN$ intersect the line $MN$ in point $L$, $N \neq L$. Prove that $ \angle BAK = \angle CAL $.

As Emily is riding her bike on a long straight road, she spots Ermenson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour. Ermenson skates at a constant rate of $8$ miles per hour. For how many minutes can Emily see Ermenson? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 16 $

In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is[i] color-identifiable[/i] if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$. For all positive integers $n$ and $t$, determine the maximum integer $g(n, t)$ such that: In any sports league with exactly $n$ distinct colors present over all teams, one can always find a color-identifiable set of size at least $g(n, t)$.

A quadrilateral $PQRS$ has an inscribed circle, the points of tangencies with sides $PQ$, $QR$, $RS$, $SP$ being $A$, $B$, $C$, $D$, respectively. Let the midpoints of $AB$, $BC$, $CD$, $DA$ be $E$, $F$, $G$, $H$, respectively. Prove that the angle between segments $PR$ and $QS$ is equal to the angle between segments $EG$ and $FH$.

Let $ABC$ be a triangle with $90^\circ \ne \angle A \ne 135^\circ$. Let $D$ and $E$ be external points to the triangle $ABC$ such that $DAB$ and $EAC$ are isoscele triangles with right angles at $D$ and $E$. Let $F = BE \cap CD$, and let $M$ and $N$ be the midpoints of $BC$ and $DE$, respectively. Prove that, if three of the points $A$, $F$, $M$, $N$ are collinear, then all four are collinear.

Consider a $2018 \times 2019$ board with integers in each unit square. Two unit squares are said to be neighbours if they share a common edge. In each turn, you choose some unit squares. Then for each chosen unit square the average of all its neighbours is calculated. Finally, after these calculations are done, the number in each chosen unit square is replaced by the corresponding average. Is it always possible to make the numbers in all squares become the same after finitely many turns?

Unitary quadratic trinomials $ f(x)$ and $ g(x)$ satisfy the following interesting condition: $ f(g(x)) \equal{} 0$ and $ g(f(x)) \equal{} 0$ do not have real roots. Prove that at least one of equations $ f(f(x)) \equal{} 0$ and $ g(g(x)) \equal{} 0$ does not have real roots too. [i]S. Berlov [/i]

Ten monkeys have 60 bananas. Each monkey has at least one banana and any two monkeys have different amounts of bananas. Prove that any six monkeys can distribute their bananas between others such that all 4 remaining monkeys have the same amount of bananas. [i]A. Voidelevich[/i]

$ABC$ is an acute triangle with orthocenter $H$. Point $P$ is in triangle $BHC$ that $\angle HPC = 3 \angle HBC $ and $\angle HPB =3 \angle HCB $. Reflection of point $P$ through $BH,CH$ is $X,Y$. if $S$ is the center of circumcircle of $AXY$ , Prove that: $$\angle BAS = \angle CAP$$ [i]Proposed by Pouria Mahmoudkhan Shirazi [/i]

A triangle $ABC$ is given. Points $E,F,G$ are arbitrarily selected on the sides $AB,BC,CA$, respectively, such that $AF\perp EG$ and the quadrilateral $AEFG$ is cyclic. Find the locus of the intersection point of $AF$ and $EG$.

Let $z$ be a complex number and k a positive integer such that $z^k$ is a positive real number other than $1$. Let $f(n)$ denote the real part of the complex number $z^n$. Assume the parabola $p(n) = an^2 +bn+c$ intersects $f(n)$ four times, at $n = 0, 1, 2, 3$. Assuming the smallest possible value of $k$, find the largest possible value of $a$.

Four circles of radius $3$ are arranged as shown. Their centers are the vertices of a square. The area of the shaded region is closest to [asy] fill((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle,lightgray); fill(arc((3,3),(0,3),(3,0),CCW)--(3,3)--cycle,white); fill(arc((3,-3),(3,0),(0,-3),CCW)--(3,-3)--cycle,white); fill(arc((-3,-3),(0,-3),(-3,0),CCW)--(-3,-3)--cycle,white); fill(arc((-3,3),(-3,0),(0,3),CCW)--(-3,3)--cycle,white); draw(circle((3,3),3)); draw(circle((3,-3),3)); draw(circle((-3,-3),3)); draw(circle((-3,3),3)); draw((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle); [/asy] $\text{(A)}\ 7.7 \qquad \text{(B)}\ 12.1 \qquad \text{(C)}\ 17.2 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 27$

Let $f:(0,\infty)\to(0,\infty)$ be a function satisfying $f\bigl(xf(y)\bigr)+f\bigl(yf(x)\bigr)=2xy$ for all $x,y>0$. Show that $f(x) = x$ for all positive $x$.

Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$