Given a triangle $ABC$ satisfying $AC+BC=3\cdot AB$. The incircle of triangle $ABC$ has center $I$ and touches the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Let $K$ and $L$ be the reflections of the points $D$ and $E$ with respect to $I$. Prove that the points $A$, $B$, $K$, $L$ lie on one circle. [i]Proposed by Dimitris Kontogiannis, Greece[/i]

George is taking a ten-question true-false exam, where the answer key has been selected uniformly at random; however, he doesn't know any of the answers! Luckily, a friend has helpfully hinted that no two consecutive questions have true as the correct answer. If George takes the exam and maximizes the expected number of questions he gets correct, how many of his answers are expected to be right?

Given $a$, $b$, and $c$ are complex numbers satisfying \[ a^2+ab+b^2=1+i \] \[ b^2+bc+c^2=-2 \] \[ c^2+ca+a^2=1, \] compute $(ab+bc+ca)^2$. (Here, $i=\sqrt{-1}$)

In the figure on the right, $ABCD$ is a con­vex quadrilateral, $K, L, M,$ and $N$ are the mid­points of its sides, and $PQRS$ is the quadrilateral formed by the intersections of $AK, BL, CM,$ and $DN$. Determine the area of quadrilateral $PQRS$ if the area of quadrilateral $ABCD$ is $3000$, and the areas of quadrilaterals $AMQP$ and $CKSR$ are $513$ and $388$, respectively. [asy] defaultpen(linewidth(0.7)+fontsize(10));size(200); pair A=origin, B=(14,0), C=(13,10), D=(2,9), K=midpoint(C--D), L=midpoint(D--A), M=midpoint(A--B), N=midpoint(B--C), P=intersectionpoint(B--L, A--K), Q=intersectionpoint(B--L, C--M), R=intersectionpoint(C--M, D--N), S=intersectionpoint(D--N, A--K); draw(K--A--B--C--D--A^^D--N^^B--L^^C--M); pair point=(7,6); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$S$", S, dir(160)*dir(point--S)); label("$R$", R, dir(190)*dir(point--R)); label("$Q$", Q, dir(180)*dir(point--Q)); label("$P$", P, dir(180)*dir(point--P)); label("$K$", K, dir(point--K)); label("$L$", L, dir(point--L)); label("$M$", M, dir(point--M)); label("$N$", N, dir(point--N));[/asy]

Let $M$ be a positive integer, and $A = \{1, 2,... , M + 1\}$. Show that if $f$ is a bijection from $A$ to $A$ then $\sum_{n=1}^{M} \frac{1}{f(n) + f(n + 1)} > \frac{M}{M + 3}$

Pingu is given two positive integers $m$ and $n$ without any common factors greater than $1$. a) Help Pingu find positive integers $p, q$ such that $$\operatorname{gcd}(pm+q, n) \cdot \operatorname{gcd}(m, pn+q) = mn$$ b) Prove to Pingu that he can never find positive integers $r, s$ such that $$\operatorname{lcm}(rm+s, n) \cdot \operatorname{lcm}(m, rn+s) = mn$$ regardless of the choice of $m$ and $n$.

Given a finite set $S \subset \mathbb{R}^3$, define $f(S)$ to be the mininum integer $k$ such that there exist $k$ planes that divide $\mathbb{R}^3$ into a set of regions, where no region contains more than one point in $S$. Suppose that \[M(n) = \max\{f(S) : |S| = n\} \text{ and } m(n) = \min\{f(S) : |S| = n\}.\] Evaluate $M(200) \cdot m(200)$.

Given two positives integers $m$ and $n$, prove that there exists a positive integer $k$ and a set $S$ of at least $m$ multiples of $n$ such that the numbers $\frac {2^k{\sigma({s})}} {s}$ are odd for every $s \in S$. $\sigma({s})$ is the sum of all positive integers of $s$ (1 and $s$ included).

Let $ABC$ be a triangle such that angle $C$ is obtuse. Let $D\in [AB]$ and $[DC]\perp [BC]$. If $m(\widehat{ABC})=\alpha$, $m(\widehat{BCA})=3\alpha$, and $|AC|-|AD|=10$, what is $|BD|$? $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 22 $

Determine the greatest power of $2$ that is a factor of $3^{15}+3^{11}+3^{6}+1$. [i]Proposed by Nathan Xiong[/i]

Let be a natural number $ n\ge 3. $ Find $$ \inf_{\stackrel{ x_1,x_2,\ldots ,x_n\in\mathbb{R}_{>0}}{1=P\left( x_1,x_2,\ldots ,x_n\right)}}\sum_{i=1}^n\left( \frac{1}{x_i} -x_i \right) , $$ where $ P\left( x_1,x_2,\ldots ,x_n\right) :=\sum_{i=1}^n \frac{1}{x_i+n-1} , $ and find in which circumstances this infimum is attained.

Let us consider a set $S = \{ a_1 < a_2 < \ldots < a_{2004}\}$, satisfying the following properties: $f(a_i) < 2003$ and $f(a_i) = f(a_j) \quad \forall i, j$ from $\{1, 2,\ldots , 2004\}$, where $f(a_i)$ denotes number of elements which are relatively prime with $a_i$. Find the least positive integer $k$ for which in every $k$-subset of $S$, having the above mentioned properties there are two distinct elements with greatest common divisor greater than 1.

Find the smallest positive integer $k$ such that $k!$ ends in at least $43$ zeroes.

For all $a,b,c\in \bb{R}^+ $ such that $a+b+c=1$ and $ ( \frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} )(a-bc)(b-ac)(c-ab)\le M \cdot abc$. Find min $M$

Let $n$ be a natural number. A sequence $x_1,x_2, \cdots ,x_{n^2}$ of $n^2$ numbers is called $n-\textit{good}$ if each $x_i$ is an element of the set $\{1,2,\cdots ,n\}$ and the ordered pairs $\left(x_i,x_{i+1}\right)$ are all different for $i=1,2,3,\cdots ,n^2$ (here we consider the subscripts modulo $n^2$). Two $n-$good sequences $x_1,x_2,\cdots ,x_{n^2}$ and $y_1,y_2,\cdots ,y_{n^2}$ are called $\textit{similar}$ if there exists an integer $k$ such that $y_i=x_{i+k}$ for all $i=1,2,\cdots,n^2$ (again taking subscripts modulo $n^2$). Suppose that there exists a non-trivial permutation (i.e., a permutation which is different from the identity permutation) $\sigma$ of $\{1,2,\cdots ,n\}$ and an $n-$ good sequence $x_1,x_2,\cdots,x_{n^2}$ which is similar to $\sigma\left(x_1\right),\sigma\left(x_2\right),\cdots ,\sigma\left(x_{n^2}\right)$. Show that $n\equiv 2\pmod{4}$.

For a permutation $\pi$ of $\{1,2,3,\ldots,n\}$, let $\text{Inv}(\pi)$ be the number of pairs $(i,j)$ with $1 \leq i < j \leq n$ and $\pi(i) > \pi(j)$. [list=1] [*] Given $n$, what is $\sum \text{Inv}(\pi)$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$? [*] Given $n$, what is $\sum \left(\text{Inv}(\pi)\right)^2$ where the sum ranges over all permutations $\pi$ of $\{1,2,3,\ldots,n\}$?[/list] [i]Brian Hamrick.[/i]

Triangle $ABC$ has $\angle{A}=90^{\circ}$, $AB=2$, and $AC=4$. Circle $\omega_1$ has center $C$ and radius $CA$, while circle $\omega_2$ has center $B$ and radius $BA$. The two circles intersect at $E$, different from point $A$. Point $M$ is on $\omega_2$ and in the interior of $ABC$, such that $BM$ is parallel to $EC$. Suppose $EM$ intersects $\omega_1$ at point $K$ and $AM$ intersects $\omega_1$ at point $Z$. What is the area of quadrilateral $ZEBK$?

Let $ABC$ be a triangle and $m$ a line which intersects the sides $AB$ and $AC$ at interior points $D$ and $F$, respectively, and intersects the line $BC$ at a point $E$ such that $C$ lies between $B$ and $E$. The parallel lines from the points $A$, $B$, $C$ to the line $m$ intersect the circumcircle of triangle $ABC$ at the points $A_1$, $B_1$ and $C_1$, respectively (apart from $A$, $B$, $C$). Prove that the lines $A_1E$ , $B_1F$ and $C_1D$ pass through the same point. [i]Greece[/i]

Incircle $\omega$ of triangle $ABC$ is tangent to sides $CB$ and $CA$ at $D$ and $E$, respectively. Point $X$ is the reflection of $D$ with respect to $B$. Suppose that the line $DE$ is tangent to the $A$-excircle at $Z$. Let the circumcircle of triangle $XZE$ intersect $\omega$ for the second time at $K$. Prove that the intersection of $BK$ and $AZ$ lies on $\omega$. Proposed by Mahdi Etesamifard and Alireza Dadgarnia

It is possible to compose a triangle from the altitudes of a given triangle. Can we conclude that it is possible to compose a triangle from its bisectors?

Let $ABCD$ be a cyclie quadrilateral, $\omega$ be it's circumcircle and $M$ be the midpoint of the arc $AB$ of $\omega$ which does not contain the vertices $C$ and $D$. The line that passes through $M$ and the intersection point of segments $AC$ and $BD$, intersects again $\omega$ in $N$. Let $P$ and $Q$ be points in the $CD$ segment such that $\angle AQD = \angle DAP$ and $\angle BPC = \angle CBQ$. Prove that the circumcircle of $NPQ$ and $\omega$ are tangent to each other.

Find all pairs $(a,b)$ of positive integers for which $\gcd(a,b)=1$, and $\frac{a}{b}=\overline{b.a}$. (For example, if $a=92$ and $b=13$, then $b/a=13.92$ )

In a tennis tournament, any two of the $n$ participants played a match (the winner of a match gets $1$ point, the loser gets $0$). The scores after the tournament were $r_1\le r_2\le\ldots\le r_n$. A match between two players is called wrong if after it the winner has a score less than or equal to that of the loser. Consider the set $I=\{i|r_1\ge i\}$. Prove that the number of wrong matches is not less than $\sum_{i\in I}(r_i-i+1)$, and show that this value is realizable

In the interior of a cube we consider $\displaystyle 2003$ points. Prove that one can divide the cube in more than $\displaystyle 2003^3$ cubes such that any point lies in the interior of one of the small cubes and not on the faces.

Suppose that $f:P(\mathbb N)\longrightarrow \mathbb N$ and $A$ is a subset of $\mathbb N$. We call $f$ $A$-predicting if the set $\{x\in \mathbb N|x\notin A, f(A\cup x)\neq x \}$ is finite. Prove that there exists a function that for every subset $A$ of natural numbers, it's $A$-predicting. [i]proposed by Sepehr Ghazi-Nezami[/i]