There is a point $D$ on side $AC$ of acute triangle $\triangle ABC$. Let $AM$ be the median drawn from $A$ (so $M$ is on $BC$) and $CH$ be the altitude drawn from $C$ (so $H$ is on $AB$). Let $I$ be the intersection of $AM$ and $CH$, and let $K$ be the intersection of $AM$ and line segment $BD$. We know that $AK=8$, $BK=8$, and $MK=6$. Find the length of $AI$.

Determine, for any positive real number $a$, the number of solutions $(x,y)$ to the system of equations $$\begin{cases} |x|+|y|= 1 \\ x^2 + y^2 = a \end{cases}$$ where $x$ and $y$ are real numbers.

A particular coin can land on heads (H), on tails (T), or in the middle (M), each with probability $\frac{1}{3}$. Find the expected number of flips necessary to observe the contiguous sequence HMMTHMMT...HMMT, where the sequence HMMT is repeated 2016 times.

Let $ABC$ be a triangle with $AB > AC$. Point $P \in (AB)$ is such that $\angle ACP = \angle ABC$. Let $D$ be the reflection of $P$ into the line $AC$ and let $E$ be the point in which the circumcircle of $BCD$ meets again the line $AC$. Prove that $AE = AC$.

Let $ S$ be the set of natural numbers $ n$ satisfying the following conditions: $ (i)$ $ n$ has $ 1000$ digits, $ (ii)$ all the digits of $ n$ are odd, and $ (iii)$ any two adjacent digits of $ n$ differ by $ 2$. Determine the number of elements of $ S$.

Evaluate \[\lim_{n\to\infty} \int_0^{\pi} x^2 |\sin nx| dx\]

Let $S$ be a finite set of points on a plane, where no three points are collinear, and the convex hull of $S$, $\Omega$, is a $2016-$gon $A_1A_2\ldots A_{2016}$. Every point on $S$ is labelled one of the four numbers $\pm 1,\pm 2$, such that for $i=1,2,\ldots , 1008,$ the numbers labelled on points $A_i$ and $A_{i+1008}$ are the negative of each other. Draw triangles whose vertices are in $S$, such that any two triangles do not have any common interior points, and the union of these triangles is $\Omega$. Prove that there must exist a triangle, where the numbers labelled on some two of its vertices are the negative of each other.

Find all positive integers $n$ such that the number $n^5+79$ has all the same digits when it is written in decimal represantation.

Let $m$ boxes be given, with some balls in each box. Let $n < m$ be a given integer. The following operation is performed: choose $n$ of the boxes and put $1$ ball in each of them. Prove: [i](a) [/i]If $m$ and $n$ are relatively prime, then it is possible, by performing the operation a finite number of times, to arrive at the situation that all the boxes contain an equal number of balls. [i](b)[/i] If $m$ and $n$ are not relatively prime, there exist initial distributions of balls in the boxes such that an equal distribution is not possible to achieve.

A circle of radius $ 1$ is internally tangent to two circles of radius $ 2$ at points $ A$ and $ B$, where $ AB$ is a diameter of the smaller circle. What is the area of the region, shaded in the gure, that is outside the smaller circle and inside each of the two larger circles? [asy]size(200);defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair B = (0,1); pair A = (0,-1); label("$B$",B,NW);label("$A$",A,2S); draw(Circle(A,2));draw(Circle(B,2)); fill((-sqrt(3),0)..B..(sqrt(3),0)--cycle,gray); fill((-sqrt(3),0)..A..(sqrt(3),0)--cycle,gray); draw((-sqrt(3),0)..B..(sqrt(3),0)); draw((-sqrt(3),0)..A..(sqrt(3),0)); path circ = Circle(origin,1); fill(circ,white); draw(circ); dot(A);dot(B); pair A1 = B + dir(45)*2; pair A2 = dir(45); pair A3 = dir(-135)*2 + A; draw(B--A1,EndArrow(HookHead,2)); draw(origin--A2,EndArrow(HookHead,2)); draw(A--A3,EndArrow(HookHead,2)); label("$2$",midpoint(B--A1),NW); label("$1$",midpoint(origin--A2),NW); label("$2$",midpoint(A--A3),NW);[/asy]$ \textbf{(A)}\ \frac {5}{3}\pi \minus{} 3\sqrt {2}\qquad \textbf{(B)}\ \frac {5}{3}\pi \minus{} 2\sqrt {3}\qquad \textbf{(C)}\ \frac {8}{3}\pi \minus{} 3\sqrt {3}\qquad\textbf{(D)}\ \frac {8}{3}\pi \minus{} 3\sqrt {2}$ $ \textbf{(E)}\ \frac {8}{3}\pi \minus{} 2\sqrt {3}$

Let $a_0$ be a positive integer and $a_{n + 1} =\sqrt{a_n^2 + 1}$, for all $n \ge 0$. 1) Prove that for all $a_0$ the sequence contains infinitely many integers and infinitely many irrational numbers. 2) Is there an $a_0$ for which $a_{2010}$ is an integer?

In an $m\times n$ rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all $(m,n)$ such that the first person has a winning strategy.

Let $S_1$ and $S_2$ be two circumferences, with centers $O_1$ and $O_2$ respectively, and secants on $M$ and $N$. The line $t$ is the common tangent to $S_1$ and $S_2$ closer to $M$. The points $A$ and $B$ are the intersection points of $t$ with $S_1$ and $S_2$, $C$ is the point such that $BC$ is a diameter of $S_2$, and $D$ the intersection point of the line $O_1O_2$ with the perpendicular line to $AM$ through $B$. Show that $M$, $D$ and $C$ are collinear.

Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$. [i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]

If $a, b, c$ are positive real numbers, prove that $$\frac{a}{\sqrt{(a + 2b)^3}}+\frac{b}{\sqrt{(b + 2c)^3}} +\frac{c} {\sqrt{(c + 2a)^3}} \ge \frac{1}{\sqrt{a + b + c}}$$ Alexandru Mihalcu

Given an angle $ \angle QBP$ and a point $ L$ outside the angle $ \angle QBP$. Draw a straight line through $ L$ meeting $ BQ$ in $ A$ and $ BP$ in $ C$ such that the triangle $ \triangle ABC$ has a given perimeter.

Let $n$ be a positive integer. In a coordinate grid, a path from $(0,0)$ to $(2n,2n)$ consists of $4n$ consecutive unit steps $(1,0)$ or $(0,1)$. Prove that the number of paths that divide the square with vertices $(0,0),(2n,0),(2n,2n),(0,2n)$ into 2 regions with even areas is $$\frac{{4n \choose 2n} + {2n \choose n}}{2}$$

If $$u=\text{cot} 22^{\circ}30’ \text{ },\text{ } v= \frac{1}{\text{sin} 22^{\circ}30’}$$ prove that $u$ satisfies a quadratic and $v$ a quartic (4th degree) equation with integral coefficients and with leading coefficients $1$.

Given an equilateral triangle $ABC$ and a point $M$ in the plane ($ABC$). Let $A', B', C'$ be respectively the symmetric through $M$ of $A, B, C$. [b]I.[/b] Prove that there exists a unique point $P$ equidistant from $A$ and $B'$, from $B$ and $C'$ and from $C$ and $A'$. [b]II.[/b] Let $D$ be the midpoint of the side $AB$. When $M$ varies ($M$ does not coincide with $D$), prove that the circumcircle of triangle $MNP$ ($N$ is the intersection of the line $DM$ and $AP$) pass through a fixed point.

Let $a_1<a_2<\cdots<a_n$ be positive integers. Some colouring of $\mathbb{Z}$ is periodic with period $t$ such that for each $x\in \mathbb{Z}$ exactly one of $x+a_1,x+a_2,\dots,x+a_n$ is coloured. Prove that $n\mid t$. [i]Andrei Radulescu-Banu[/i]

Let $\mathcal{A}$ be the set of finite sequences of positive integers $a_1,a_2,\dots,a_k$ such that $|a_n-a_{n-1}|=a_{n-2}$ for all $3\leqslant n\leqslant k$. If $a_1=a_2=1$, and $k=18$, determine the number of elements of $\mathcal{A}$.

[b]a)[/b] Show that $ n^2+2n+2007 $ is squarefree for any natural number $ n. $ [b]b)[/b] Prove that for any natural number $ k\ge 2 $ there is a nonnegative integer $ m $ such that $ m^2+2m+2k $ is a perfect square.

In $\triangle A B C$, let $D, E$, and $F$ be the midpoints of the sides of the triangle, and let $P, Q,$ and $R$ be the midpoints of the corresponding medians, $AD ,B E,$ and $C F$, respectively, as shown in the figure at the right. Prove that the value of \[\frac{AQ^2 + A R^2 + B P^2 + B R^2 + C P^2+ C Q^2 }{A B^2 + B C^2 + C A^2}\] does not depend on the shape of $\triangle A B C$ and find that value. [asy] defaultpen(linewidth(0.7)+fontsize(10));size(200); pair A=origin, B=(14,0), C=(9,12), D=midpoint(C--B), E=midpoint(C--A), F=midpoint(A--B), R=midpoint(C--F), P=midpoint(D--A), Q=midpoint(E--B); draw(A--B--C--A, linewidth(1)); draw(A--D^^B--E^^C--F); draw(B--R--A--Q--C--P--cycle, dashed); pair point=centroid(A,B,C); 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("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$P$", P, dir(40)*dir(point--P)); label("$Q$", Q, dir(40)*dir(point--Q)); label("$R$", R, dir(40)*dir(point--R)); dot(P^^Q^^R);[/asy]

Given a polynomial $P(x)=x^4+x^3+3x^2-6x+1$. Calculate $P(\alpha^2+\alpha+1)$ where \[ \alpha=\sqrt[3]{\frac{1+\sqrt{5}}{2}}+\sqrt[3]{\frac{1-\sqrt{5}}{2}} \]

Let $ABC$ be a triangle with semiperimeter $s$ and inradius $r$. The semicircles with diameters $BC, CA, AB$ are drawn on the outside of the triangle $ABC$. The circle tangent to all three semicircles has radius $t$. Prove that $$\frac{s}{2} < t \le \frac{s}{2} + \left( 1 - \frac{\sqrt3}{2} \right)r.$$