Let $M$ be the midpoint of the side $BC$ of triangle $ABC$. The bisector of the exterior angle of point $A$ intersects the side $BC$ in $D$. Let the circumcircle of triangle $ADM$ intersect the lines $AB$ and $AC$ in $E$ and $F$ respectively. If the midpoint of $EF$ is $N$, prove that $MN\parallel AD$.

For all $n\ge2$, let $a_n$ be the product of all coprime natural numbers less than $n$. Prove that (a) $n\mid a_n+1\Leftrightarrow n=2,4,p^\alpha,2p^\alpha$ (b) $n\mid a_n-1\Leftrightarrow n\ne2,4,p^\alpha,2p^\alpha$ Here $p$ is an odd prime number and $\alpha\in\mathbb N$.

Let $P$ be a point outside of the triangle $ABC$ in the plane of $ABC$. Prove that by using reflections $S_{AB}$, $S_{AC}$, and $S_{BC}$ across the lines $AB$, $AC$, and $BC$ one can shift point $P$ inside the triangle $ABC$.

Let $ a,b,c>0$. Prove that $ \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq 3\sqrt{\frac{a^2+b^2+c^2}{ab+bc+ca}}$

Given a $n\times n$ table with non-negative real entries such that the sums of entries in each column and row are equal, a player plays the following game: The step of the game consists of choosing $n$ cells, no two of which share a column or a row, and subtracting the same number from each of the entries of the $n$ cells, provided that the resulting table has all non-negative entries. Prove that the player can change all entries to zeros.

Given a quadrilateral $ABCD$ simultaneously inscribed and circumscribed, assume that none of its diagonals or sides is a diameter of the circumscribed circle. Let $P$ be the intersection point of the external bisectors of the angles near $A$ and $B$. Similarly, let $Q$ be the intersection point of the external bisectors of the angles $C$ and $D$. If $J$ and $O$ respectively are the incenter and circumcenter of $ABCD$ prove that $OJ\perp PQ$.

Find the number of subsets of the set $\{1,2,3,...,5n\}$ such that the sum of the elements in each subset are divisible by $5$.

Given $101$ segments in a line, prove that there exists $11$ segments meeting in $1$ point or $11$ segments such that every two of them are disjoint.

Let $p$ be an odd prime number. Let $g$ be a primitive root of unity modulo $p$. Find all the values of $p$ such that the sets $A=\left\{k^2+1:1\le k\le\frac{p-1}2\right\}$ and $B=\left\{g^m:1\le m\le\frac{p-1}2\right\}$ are equal modulo $p$.

If $x,y,z\in\mathbb N$ and $xy=z^2+1$ prove that there exists integers $a,b,c,d$ such that $x=a^2+b^2$, $y=c^2+d^2$, $z=ac+bd$.

Given a point $P$ in the circumcircle $\omega$ of an equilateral triangle $ABC$, prove that the segments $PA$, $PB$, and $PC$ form a triangle $T$. Let $R$ be the radius of the circumcircle $\omega$ and let $d$ be the distance between $P$ and the circumcenter. Find the area of $T$.

Let $n=p_1^{\alpha_1}\cdots p_s^{\alpha_s}\ge2$. If for any $\alpha\in\mathbb N$, $p_i-1\nmid\alpha$, where $i=1,2,\ldots,s$, prove that $n\mid\sum_{\alpha\in\mathbb Z^*_n}\alpha^{\alpha}$ where $\mathbb Z^*_n=\{a\in\mathbb Z_n:\gcd(a,n)=1\}$.