The incircle of a non-isosceles triangle $ABC$ has centre $I$ and is tangent to $BC$ and $CA$ in $D$ and $E$, respectively. Let $H$ be the orthocentre of $ABI$, let $K$ be the intersection of $AI$ and $BH$ and let $L$ be the intersection of $BI$ and $AH$. Show that the circumcircles of $DKH$ and $ELH$ intersect on the incircle of $ABC$.

Let $E$ be a point on the median $AD$ of a triangle $ABC$, and $F$ be the projection of $E$ onto $BC$. From a point $M$ on $EF$ the perpendiculars $MN$ to $AC$ and $MP$ to $AB$ are drawn. Prove that if the points $N,E,P$ lie on a line, then $M$ lies on the bisector of $\angle BAC$.

$p,q,r$ are real numbers satisfying \[\frac{(p+q)(q+r)(r+p)}{pqr} = 24\] \[\frac{(p-2q)(q-2r)(r-2p)}{pqr} = 10.\] Given that $\frac{p}{q} + \frac{q}{r} + \frac{r}{p}$ can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers, compute $m+n$. [i]Author: Alex Zhu[/i]

A table consisting of $5$ columns and $32$ rows, which are filled with zero and one numbers, are "varied", if no two lines are filled in the same way.\\ On the exterior of a cylinder, a table with $32$ rows and $16$ columns is constructed. Is it possible to fill the numbers cells of the table with numbers zero and one, such that any five consecutive columns, table $32\times5$ created by these columns, is a varied one? [i]Proposed by Morteza Saghafian[/i]

Let $n$, $m$, and $k$ be positive integers satisfying $(n - 1)n(n + 1) = m^k$. Prove that $k = 1$.

The triangle $ABC$ is isosceles with $AB=BC$. The point F on the side $[BC]$ and the point $D$ on the side $AC$ are the feets of the the internals bisectors drawn from $A$ and altitude drawn from $B$ respectively so that $AF=2BD$. Fine the measure of the angle $ABC$.

Vertices and Edges of a regular $n$-gon are numbered $1,2,\dots,n$ clockwise such that edge $i$ lies between vertices $i,i+1 \mod n$. Now non-negative integers $(e_1,e_2,\dots,e_n)$ are assigned to corresponding edges and non-negative integers $(k_1,k_2,\dots,k_n)$ are assigned to corresponding vertices such that: $i$) $(e_1,e_2,\dots,e_n)$ is a permutation of $(k_1,k_2,\dots,k_n)$. $ii$) $k_i=|e_{i+1}-e_i|$ indexes$\mod n$. a) Prove that for all $n\geq 3$ such non-zero $n$-tuples exist. b) Determine for each $m$ the smallest positive integer $n$ such that there is an $n$-tuples stisfying the above conditions and also $\{e_1,e_2,\dots,e_n\}$ contains all $0,1,2,\dots m$.

Let $P$ be a polynomial with real coefficients. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a real number $t$ such that \[f(x+t) - f(x) = P(x)\] for all $x \in \mathbb{R}$.

Compute the positive real number $x$ satisfying $$x^{(2x^6)}=3.$$

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

In how many ways can 10001 be written as the sum of two primes? $ \textbf{(A)}0\qquad\textbf{(B)}1\qquad\textbf{(C)}2\qquad\textbf{(D)}3\qquad\textbf{(E)}4 $

There is an $n*n$ table with some unit cells colored black and the others are white. In each step , Amin takes a $row$ with exactly one black cell in it , and color all cells in that black cell's $column$ red. While Ali , takes a $column$ with exactly one black cell in it , and color all cells in that black cell's $row$ red. Prove that Amin can color all the cells red , iff Ali can do so.

Sides $AB$ and $CD$ of quadrilateral $ABCD$ intersect at point $E$. On the diagonals$ AC$ and $BD$ points $M$ and $N$ are taken, respectively, so that $AM / AC = BN / BD = k$. Find the area of ​​a triangle $EMN$ if the area of ​​$ABCD$ is $S$.

Let $R$ be a set of exactly $6$ elements. A set $F$ of subsets of $R$ is called an $S$-family over $R$ if and only if it satisfies the following three conditions: (i) For no two sets $X, Y$ in $F$ is $X \subseteq Y$ ; (ii) For any three sets $X, Y,Z$ in $F$, $X \cup Y \cup Z \neq R,$ (iii) $\bigcup_{X \in F} X = R$

Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? $ \textbf{(A) }60 \qquad \textbf{(B) }65 \qquad \textbf{(C) }70 \qquad \textbf{(D) }75 \qquad \textbf{(E) }80 \qquad $

Consider the following operation on positive real numbers written on a blackboard: Choose a number $ r$ written on the blackboard, erase that number, and then write a pair of positive real numbers $ a$ and $ b$ satisfying the condition $ 2 r^2 \equal{} ab$ on the board. Assume that you start out with just one positive real number $ r$ on the blackboard, and apply this operation $ k^2 \minus{} 1$ times to end up with $ k^2$ positive real numbers, not necessarily distinct. Show that there exists a number on the board which does not exceed kr.

Let set $ T \equal{} \{1,2,3,4,5,6,7,8\}$. Find the number of all nonempty subsets $ A$ of $ T$ such that $ 3|S(A)$ and $ 5\nmid S(A)$, where $ S(A)$ is the sum of all the elements in $ A$.

It is known that circle $\Gamma_1(O_1)$ has center at $O_1$, circle $\Gamma_2(O_2)$ has center at $O_2$, and both intersect at points $C$ and $D$. It is also known that points $P$ and $Q$ lie on circles $\Gamma_1(O_1)$ and $\Gamma_2(O_2)$, respectively. ). A line $\ell$ passes through point $D$ and intersects $\Gamma_1(O_1)$ and $\Gamma_2(O_2)$ at points $A$ and $B$, respectively. The lines $PD$ and $AC$ meet at point $M$, and the lines $QD$ and $BC$ meet at point $N$. Let $O$ be center outer circle of triangle $ABC$. Prove that $OD$ is perpendicular to $MN$ if and only if a circle can be found which passes through the points $P, Q, M$ and $N$.

It is known that a circle can be inscribed in the quadrilateral $ABCD$, in addition $\angle A = \angle C$. Prove that $AB = BC$, $CD = DA$. (Olena Artemchuk)

Let $a_1, a_2,\cdots , a_n$ and $b_1, b_2,\cdots , b_n$ be (not necessarily distinct) positive integers. We continue the sequences as follows: For every $i>n$, $a_i$ is the smallest positive integer which is not among $b_1, b_2,\cdots , b_{i-1}$, and $b_i$ is the smallest positive integer which is not among $a_1, a_2,\cdots , a_{i-1}$. Prove that there exists $N$ such that for every $i>N$ we have $a_i=b_i$ or for every $i>N$ we have $a_{i+1}=a_i$.

A strange calculator has only two buttons with positive itegers, each of them consisting of two digits. It displays the number 1 at the beginning. Whenever a button with number $N$ is pressed, the calculator replaces the displayed number $X$ with the number $X\cdot N$ or $X+N$. Multiplication and addition alternate, multiplication is the first. (For example,if the number 10 is on the 1st button, the number 20 is on the 2nd button, and we consecutively press the 1st, 2nd, 1st and 1st button, we get the results $1\cdot 10=10$, $10+20=30$, $30\cdot 10=300$, and $300+10=310$.) Decide whether there exist particular values of the two-digit nubers on the buttons such that one can display infinitely many numbers (without cleaning the display, i.e. you must keep going and get infinitel many numbers) ending with (a) $2015$, (b) $5813$. [i]Proposed by Michal Rolínek and Peter Novotný[/i]

Acute-angled triangle $ABC$ is inscribed in a circle with center at $O$; $\stackrel \frown {AB} = 120$ and $\stackrel \frown {BC} = 72$. A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. Then the ratio of the magnitudes of angles $OBE$ and $BAC$ is: $\textbf{(A)}\ \dfrac{5}{18} \qquad \textbf{(B)}\ \dfrac{2}{9} \qquad \textbf{(C)}\ \dfrac{1}{4} \qquad \textbf{(D)}\ \dfrac{1}{3} \qquad \textbf{(E)}\ \dfrac{4}{9}$

$3$ builders are scheduled to build a house in $60$ days. However, they suffer from a bout of procrastination and thus do nothing for the first $50$ days. Panicked, they realize in order to build the house on time, they must hire more workers [i]and[/i] work twice as fast as they would have originally. If the new workers they hire also will work at the doubled rate, how many new workers will they need to hire? Assume each builder works at the same rate as the others and they do not get in each other's way.

The vertices of a regular nonagon are colored such that $1)$ adjacent vertices are different colors and $2)$ if $3$ vertices form an equilateral triangle, they are all different colors. Let $m$ be the minimum number of colors needed for a valid coloring, and n be the total number of colorings using $m$ colors. Determine $mn$. (Assume each vertex is distinguishable.)

For which real numbers $a$ does the sequence $(u_n )$ defined by the initial condition $u_0 =a$ and the recursion $u_{n+1} =2u_n - n^2$ have $u_n >0$ for all $n \geq 0?$