Three circles $\omega_1,\omega_2,\omega_3$ are tangent to line $l$ at points $A,B,C$ ($B$ lies between $A,C$) and $\omega_2$ is externally tangent to the other two. Let $X,Y$ be the intersection points of $\omega_2$ with the other common external tangent of $\omega_1,\omega_3$. The perpendicular line through $B$ to $l$ meets $\omega_2$ again at $Z$. Prove that the circle with diameter $AC$ touches $ZX,ZY$. [i]Proposed by Iman Maghsoudi - Siamak Ahmadpour[/i]

Compute $$ \lim_{A\to+\infty}\frac1A\int_1^A A^{\frac1x}\, dx . $$ Proposed by Jan Šustek, University of Ostrava

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$ and circumcentre $O$. Points $D\neq B$ and $E\neq C$ lie on $\omega$ such that $BD\perp AC$ and $CE\perp AB$. Let $CO$ meet $AB$ at $X$, and $BO$ meet $AC$ at $Y$. Prove that the circumcircles of triangles $BXD$ and $CYE$ have an intersection lie on line $AO$. [i]Ivan Chan Kai Chin, Malaysia[/i]

Let $n \geq 2$ be a positive integer. The set $M$ consists of $2n^2-3n+2$ positive rational numbers. Prove that there exists a subset $A$ of $M$ with $n$ elements with the following property: $\forall$ $2 \leq k \leq n$ the sum of any $k$ (not necessarily distinct) numbers from $A$ is not in $A$.

Consider this histogram of the scores for $81$ students taking a test: [asy] unitsize(12); draw((0,0)--(26,0)); draw((1,1)--(25,1)); draw((3,2)--(25,2)); draw((5,3)--(23,3)); draw((5,4)--(21,4)); draw((7,5)--(21,5)); draw((9,6)--(21,6)); draw((11,7)--(19,7)); draw((11,8)--(19,8)); draw((11,9)--(19,9)); draw((11,10)--(19,10)); draw((13,11)--(19,11)); draw((13,12)--(19,12)); draw((13,13)--(17,13)); draw((13,14)--(17,14)); draw((15,15)--(17,15)); draw((15,16)--(17,16)); draw((1,0)--(1,1)); draw((3,0)--(3,2)); draw((5,0)--(5,4)); draw((7,0)--(7,5)); draw((9,0)--(9,6)); draw((11,0)--(11,10)); draw((13,0)--(13,14)); draw((15,0)--(15,16)); draw((17,0)--(17,16)); draw((19,0)--(19,12)); draw((21,0)--(21,6)); draw((23,0)--(23,3)); draw((25,0)--(25,2)); for (int a = 1; a < 13; ++a) { draw((2*a,-.25)--(2*a,.25)); } label("$40$",(2,-.25),S); label("$45$",(4,-.25),S); label("$50$",(6,-.25),S); label("$55$",(8,-.25),S); label("$60$",(10,-.25),S); label("$65$",(12,-.25),S); label("$70$",(14,-.25),S); label("$75$",(16,-.25),S); label("$80$",(18,-.25),S); label("$85$",(20,-.25),S); label("$90$",(22,-.25),S); label("$95$",(24,-.25),S); label("$1$",(2,1),N); label("$2$",(4,2),N); label("$4$",(6,4),N); label("$5$",(8,5),N); label("$6$",(10,6),N); label("$10$",(12,10),N); label("$14$",(14,14),N); label("$16$",(16,16),N); label("$12$",(18,12),N); label("$6$",(20,6),N); label("$3$",(22,3),N); label("$2$",(24,2),N); label("Number",(4,8),N); label("of Students",(4,7),N); label("$\textbf{STUDENT TEST SCORES}$",(14,18),N); [/asy] The median is in the interval labeled $\text{(A)}\ 60 \qquad \text{(B)}\ 65 \qquad \text{(C)}\ 70 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 80$

Let $ABC$ be an acute-angled triangle with $AC<BC.$ A circle passes through $A$ and $B$ and crosses the segments $AC$ and $BC$ again at $A_1$ and $B_1$ respectively. The circumcircles of $A_1B_1C$ and $ABC$ meet each other at points $P$ and $C.$ The segments $AB_1$ and $A_1B$ intersect at $S.$ Let $Q$ and $R$ be the reflections of $S$ in the lines $CA$ and $CB$ respectively. Prove that the points $P,$ $Q,$ $R,$ and $C$ are concyclic.

Let $R=(8,6)$. The lines whose equations are $8y=15x$ and $10y=3x$ contain points $P$ and $Q$, respectively, such that $R$ is the midpoint of $\overline{PQ}$. The length of $PQ$ equals $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

If the square of a number of two digits is decreased by the square of the number formed by reversing the digits, then the result is not always divisible by: $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ \text{the product of the digits}\qquad \textbf{(C)}\ \text{the sum of the digits}\qquad \textbf{(D)}\ \text{the difference of the digits}\qquad \textbf{(E)}\ 11$

In a pentagon $\Pi$ in the plane, $M_1,...M_5$ are the midpoints of the consecutive sides. $Z_i$ is the centroid of the triangle $M_{i} M_{i+1} M_{i+3}$, where $i=1,2...5$ and it is understood that $M_{j\cdot 5}=M_j$ Given pentagon $Z_{1}Z_{2}Z_{3}Z_{4}Z_{5}$, determine the original pentagon $\Pi$.

Which integers can be represented as \[\frac{(x+y+z)^{2}}{xyz}\] where $x$, $y$, and $z$ are positive integers?

$A_1,A_2,B_1,B_2,C_1,C_2$ are points on a circle such that $A_1A_2 \parallel B_1B_2 \parallel C_1C_2 $ . $M$ is a point on same circle $MA_1$ and $B_2C_2$ intersect at $X$ , $MB_1$ and $A_2C_2$ intersect at $Y$, $MC_1$ and $A_2B_2$ intersect at $Z$ .Prove that $X , Y ,Z$ are collinear.

Find all tuples of positive integers $(a,b,c)$ such that $$a^b+b^c+c^a=a^c+b^a+c^b$$

Find all pairs of integers $x,y$ for which \[x^3+x^2+x=y^2+y.\]

Determine continuous functions $f:\mathbb{R}\to \mathbb{R}$ such that $\left( {{a}^{2}}+ab+{{b}^{2}} \right)\int\limits_{a}^{b}{f\left( x \right)dx=3\int\limits_{a}^{b}{{{x}^{2}}f\left( x \right)dx,}}$ for every $a,b\in \mathbb{R}$ .

Evaluate $\int_1^e \frac{\ln x-1}{x^2-(\ln x)^2}dx.$

Consider integers $m\ge 2$ and $n\ge 1$. Show that there is a polynomial $P(x)$ of degree equal to $n$ with integer coefficients such that $P(0),P(1),...,P(n)$ are all perfect powers of $m$ .

An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$, such that: $(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$; $(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$. Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.

The integers $1, 2, \dots, 64$ are written in the squares of a $8 \times 8$ chess board, such that for each $1 \le i < 64$, the numbers $i$ and $i+1$ are in squares that share an edge. What is the largest possible sum that can appear along one of the diagonals?

Given a cyclic quadriteral $ABCD$. The circumcenter lies in the quadriteral $ABCD$. Diagonals $AC$ and $BD$ intersects at $S$. Points $P$ and $Q$ are the midpoints of $AD$ and $BC$. Let $p$ be a line perpendicular to $AC$ through $P$, $q$ perpendicular line to $BD$ through $Q$ and $s$ perpendicular to $CD$ through $S$. Prove that $p,q,s$ intersects at one point.

A four-pointed star is formed by placing for equilateral triangles of side length $4$ in a coordinate grid. The triangles are placed such that their bases lie along one of the coordinate axes, with the midpoint of the bases lying at the origin, and such that the vertices opposite the bases lie at four distinct points. Compute the area contained within the star.

Let $S$ be the set of all partitions of $2000$ (in a sum of positive integers). For every such partition $p$, we define $f (p)$ to be the sum of the number of summands in $p$ and the maximal summand in $p$. Compute the minimum of $f (p)$ when $p \in S .$

Given a set $S \subset N$ and a positive integer n, let $S\oplus \{n\} = \{s+n / s \in S\}$. The sequence $S_k$ of sets is defined inductively as follows: $S_1 = {1}$, $S_k=(S_{k-1} \oplus \{k\}) \cup \{2k-1\}$ for $k = 2,3,4, ...$ (a) Determine $N - \cup _{k=1}^{\infty} S_k$. (b) Find all $n$ for which $1994 \in S_n$.

Let $A_1,B_1,C_1$ be the midpoints of the sides $BC,CA,AB$ respectively of a triangle $ABC$. Points $K$ on segment $C_1A_1$ and $L$ on segment $A_1B_1$ are taken such that $$\frac{C_1K}{KA_1}=\frac{BC+AC}{AC+AB}\enspace\enspace\text{and}\enspace\enspace\frac{A_1L}{LB_1}=\frac{AC+AB}{BC+AB}.$$If $BK$ and $CL$ meet at $S$, prove that $\angle C_1A_1S=\angle B_1A_1S$.

Let $P_1, P_2,..., P_n$ be a convex $n$-gon. If all lines $P_iP_j$ are joined, what is the maximum possible number of intersections in terms of $n$ obtained from strictly inside the polygon?

Let $S$ be a subset of the set $\{1, 2, 3, \dots, 2015\}$ such that for any two elements $a, b \in S$, the difference $a - b$ does not divide the sum $a + b$. Find the maximum possible size of $S$.