From the foot of one altitude of the acute triangle, perpendiculars are drawn on the other two sides, that meet the other sides at $P$ and $Q$. Show that the length of $PQ$ does not depend on which of the three altitudes is selected.

Harry wants to label the points of a regular octagon with numbers $1,2,\ldots ,8$ and label the edges with $1,2,\ldots, 8$. There are special rules he must follow: If an edge is numbered even, then the sum of the numbers of its endpoints must also be even. If an edge is numbered odd, then the sum of the numbers of its endpoints must also be odd. Two octagon labelings are equivalent if they can be made equal up to rotation, but not up to reflection. If $N$ is the number of possible octagon labelings, find the remainder when $N$ is divided by $100$. [i]Proposed by Harry Kim[/i]

Consider a circle with center $O$ and radius $R,$ and let $A$ and $B$ be two points in the plane of this circle. [b]a.)[/b] Draw a chord $CD$ of the circle such that $CD$ is parallel to $AB,$ and the point of the intersection $P$ of the lines $AC$ and $BD$ lies on the circle. [b]b.)[/b] Show that generally, one gets two possible points $P$ ($P_{1}$ and $P_{2}$) satisfying the condition of the above problem, and compute the distance between these two points, if the lengths $OA=a,$ $OB=b$ and $AB=d$ are given.

Find all sets $ A$ of nonnegative integers with the property: if for the nonnegative intergers $m$ and $ n $ we have $m+n\in A$ then $m\cdot n\in A.$

Ted is five times as old as Rosie was when Ted was Rosie's age. When Rosie reaches Ted's current age, the sum of their ages will be $72$. Find Ted's current age.

Define $\{p_n\}_{n=0}^\infty\subset\mathbb N$ and $\{q_n\}_{n=0}^\infty\subset\mathbb N$ to be sequences of natural numbers as follows: [list] [*]$p_0=q_0=1$; [*]For all $n\in\mathbb N$, $q_n$ is the smallest natural number such that there exists a natural number $p_n$ with $\gcd(p_n,q_n)=1$ satisfying \[\dfrac{p_{n-1}}{q_{n-1}} < \dfrac{p_n}{q_n} < \sqrt 2.\] [/list] Find $q_3$.

A pair of positive integers $m, n$ is called [i]guerrera[/i], if there exists positive integers $a, b, c, d$ such that $m=ab$, $n=cd$ and $a+b=c+d$. For example the pair $8, 9$ is [i]guerrera[/i] cause $8= 4 \cdot 2$, $9= 3 \cdot 3$ and $4+2=3+3$. We paint the positive integers if the following order: We start painting the numbers $3$ and $5$. If a positive integer $x$ is not painted and a positive $y$ is painted such that the pair $x, y$ is [i]guerrera[/i], we paint $x$. Find all positive integers $x$ that can be painted.

Find all sequences $a_{0}, a_{1},\ldots, a_{n}$ of real numbers such that $a_{n}\neq 0$, for which the following statement is true: If $f: \mathbb{R}\to\mathbb{R}$ is an $n$ times differentiable function and $x_{0}<x_{1}<\ldots <x_{n}$ are real numbers such that $f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0$ then there is $h\in (x_{0}, x_{n})$ for which \[a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.\]

Let $n$ be a positive integer and let $x_1\le x_2\le\cdots\le x_n$ be real numbers. Prove that \[ \left(\sum_{i,j=1}^{n}|x_i-x_j|\right)^2\le\frac{2(n^2-1)}{3}\sum_{i,j=1}^{n}(x_i-x_j)^2. \] Show that the equality holds if and only if $x_1, \ldots, x_n$ is an arithmetic sequence.

The average age of the 40 members of a computer science camp is 17 years. There are 20 girls, 15 boys, and 5 adults. If the average age of the girls is 15 and the average age of the boys is 16, what is the average age of the adults? $ \text{(A)}\ 26\qquad\text{(B)}\ 27\qquad\text{(C)}\ 28\qquad\text{(D)}\ 29\qquad\text{(E)}\ 30 $

Given a succession $C$ of $1001$ positive real numbers (not necessarily distinct), and given a set $K$ of distinct positive integers, the permitted operation is: select a number $k\in{K}$, then select $k$ numbers in $C$, calculate the arithmetic mean of those $k$ numbers, and replace each of those $k$ selected numbers with the mean. If $K$ is a set such that for each $C$ we can reach, by a sequence of permitted operations, a state where all the numbers are equal, determine the smallest possible value of the maximum element of $K$.

What is the degree measure of the acute angle formed by lines with slopes $2$ and $\tfrac{1}{3}$? $\textbf{(A)}~30\qquad\textbf{(B)}~37.5\qquad\textbf{(C)}~45\qquad\textbf{(D)}~52.5\qquad\textbf{(E)}~60$

Find the number of ways in which three numbers can be selected from the set $\{1,2,\cdots ,4n\}$, such that the sum of the three selected numbers is divisible by $4$.

In a quadrilateral $ABCD$, we have $\angle DAB = 110^{\circ} , \angle ABC = 50^{\circ}$ and $\angle BCD = 70^{\circ}$ . Let $ M, N$ be the mid-points of $AB$ and $CD$ respectively. Suppose $P$ is a point on the segment $M N$ such that $\frac{AM}{CN} = \frac{MP}{PN}$ and $AP = CP$ . Find $\angle AP C$.

Let $AA', BB'$ and $CC'$ be the altitudes of an acute-angled triangle $ABC$. Points $C_a, C_b$ are symmetric to $C' $ wrt $AA'$ and $BB'$. Points $A_b, A_c, B_c, B_a$ are defined similarly. Prove that lines $A_bB_a, B_cC_b$ and $C_aA_c$ are parallel.

Determine all integral solutions of \[ a^2\plus{}b^2\plus{}c^2\equal{}a^2b^2.\]

Let $x,y$ and $z$ be positive real numbers. Show that $x^2+xy^2+xyz^2\ge 4xyz-4$.

Fill in the grid below with the numbers $1$ through $25$, with each number used exactly once, subject to the following constraints: [list=1] [*] Each shaded square contains an even number, and each unshaded square contains an odd number. [/*] [*] For any pair of squares that share a side, if $x$ and $y$ are the two numbers in those squares, then either $x\geq2y$ or $y\geq2x$. [/*] [/list] Four numbers have been filled in already. [asy] size(10cm); for(int i=1; i<5; ++i){ draw((-2i+1,-2i+9)--(2i-1,-2i+9)); draw((-2i+1,2i-9)--(2i-1,2i-9)); draw((-2i+9,-2i+1)--(-2i+9,2i-1)); draw((2i-9,-2i+1)--(2i-9,2i-1)); } for(int i=1; i<3; ++i){ filldraw((-1,2i+1)--(-1,2i-1)--(-3,2i-1)--(-3,2i+1)--cycle,lightgray); } for(int i=2; i<4; ++i){ filldraw((1,2i+1)--(1,2i-1)--(-1,2i-1)--(-1,2i+1)--cycle,lightgray); } for(int i=1; i<5; ++i){ filldraw((3,2i-5)--(3,2i-7)--(1,2i-7)--(1,2i-5)--cycle,lightgray); } filldraw((-5,1)--(-5,-1)--(-7,-1)--(-7,1)--cycle,lightgray); filldraw((-1,-1)--(-1,-3)--(-3,-3)--(-3,-1)--cycle,lightgray); filldraw((1,-5)--(1,-7)--(-1,-7)--(-1,-5)--cycle,lightgray); filldraw((5,3)--(5,1)--(3,1)--(3,3)--cycle,lightgray); label("\Huge{25}",(-4,2)); label("\Huge{13}",(0,0)); label("\Huge{16}",(0,6)); label("\Huge{21}",(4,-2)); [/asy]

Prove that for any integer $n$ greater than $5$, a square can be divided into $n$ squares.

The sequence $a_1,a_2,a_3,...$, is constructed according to the rule $$a_1=1, a_2=a_1+1/a_1, ... , a_{n+1}=a_n+1/a_n, ...$$ Prove that $a_{100} > 14$.

Let $m$ and $n$ be positive integers greater than $1$. In each unit square of an $m\times n$ grid lies a coin with its tail side up. A [i]move[/i] consists of the following steps. [list=1] [*]select a $2\times 2$ square in the grid; [*]flip the coins in the top-left and bottom-right unit squares; [*]flip the coin in either the top-right or bottom-left unit square. [/list] Determine all pairs $(m,n)$ for which it is possible that every coin shows head-side up after a finite number of moves. [i]Thanasin Nampaisarn, Thailand[/i]

Let $a, b,$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove the following inequality $$a \sqrt[3]{\frac{b}{a}} + b \sqrt[3]{\frac{c}{b}} + c \sqrt[3]{\frac{a}{c}} \le ab + bc + ca + \frac{2}{3}.$$ Proposed by [i]Anastasija Trajanova, Macedonia[/i]

Write 11 numbers on a sheet of paper six zeros and five ones. Perform the following operation 10 times: cross out any two numbers, and if they were equal, write another zero on the board. If they were not equal, write a one. Show that no matter which numbers are chosen at each step, the nal number on the board will be a one.

It is known that $1$ and $2$ are roots of a polynomial with integer coefficients. Prove that the polynomial has a coefficient with value less than $-1$ .

What is the smalles positive integer with exactly $768$ divisors? Your answer may be written in its prime factorization.