Points $M$ and $N$ lie on the sides $BC$ and $CD$ of the square $ABCD,$ respectively, and $\angle MAN = 45^{\circ}$. The circle through $A,B,C,D$ intersects $AM$ and $AN$ again at $P$ and $Q$, respectively. Prove that $MN || PQ.$

Find the maximum possible value of $a + b + c ,$ if $a,b,c$ are positive real numbers such that $a^2 + b^2 + c^2 = a^3 + b^3 + c^3 .$

Determine the last four digits of a perfect square of a natural number, knowing that the last three of them are the same.

Removing a unit square from a $2\times 2$ square we get a piece called L-tromino. From the fourth line of a $7 \times 7$ cheesboard some unit squares have been removed. The resulting chessboard is cut in L-trominos. Determine the number and location of the removed squares.

From the foot $D$ of the height $CD$ in the triangle $ABC,$ perpendiculars to $BC$ and $AC$ are drawn, which they intersect at points $M$ and $N.$ Let $\angle CAB = 60^{\circ} , \angle CBA = 45^{\circ} ,$ and $H$ be the orthocentre of $MNC.$ If $O$ is the midpoint of $CD,$ find $\angle COH.$

If $a,b$ are real numbers such that $a^3 +12a^2 + 49a + 69 = 0$ and $ b^3 - 9b^2 + 28b - 31 = 0,$ find $a + b .$

A $9\times 1$ rectangle is divided into unit squares. A broken line, from the lower left to the upper right corner, goes through all $20$ vertices of the unit squares and consists of $19$ line segments. How many such lines are there?

Find the smallest positive integer $n,$ such that $3^k+n^k+ (3n)^k+ 2014^k$ is a perfect square for all natural numbers $k,$ but not a perfect cube, for all natural numbers $k.$