The paper is written on consecutive integers $1$ through $n$. Then are deleted all numbers ending in $4$ and $9$ and the rest alternating between $-$ and $+$. Finally, an opening parenthesis is added after each character and at the end of the expression the corresponding number of parentheses: $1 - (2 + 3 - (5 + 6 - (7 + 8 - (10 +...))))$. Find all numbers $n$ such that the value of this expression is $13$.

A farmer noticed that, during the last year, there were exactly as many calves born as during the two preceding years together. Even better, the number of pigs born during the last year was one larger than the number of pigs born during the two preceding years together. The farmer promised that if such a trend will continue then, after some years, at least twice as many pigs as calves will be born in his cattle, even though this far this target has not yet ever been reached. Will the farmer be able to keep his promise?

Let ABCD be a parallelogram, M the midpoint of AB and N the intersection of CD and the angle bisector of ABC. Prove that CM and BN are perpendicular iff AN is the angle bisector of DAB.

Does there exist a natural number with the sum of digits of its $ kth$ power being equal to $ k$, if a) $ k = 2004$; b) $ k = 2006?$

A $ 9 \times 9$ square is divided into unit squares. Is it possible to fill each unit square with a number $ 1, 2,..., 9$ in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?

Find all real numbers with the following property: the difference of its cube and its square is equal to the square of the difference of its square and the number itself.

A solid figure consisting of unit cubes is shown in the picture. Is it possible to exactly fill a cube with these figures if the side length of the cube is a) 15; b) 30?

Two non-intersecting circles, not lying inside each other, are drawn in the plane. Two lines pass through a point $P$ which lies outside each circle. The first line intersects the first circle at $A$ and $A'$ and the second circle at $B$ and $B'$, here $A$ and $B$ are closer to $P$ than $A'$ and $B'$, respectively, and $P$ lies on segment $AB$. Analogously, the second line intersects the first circle at $C$ and $C'$ and the second circle at $D$ and $D'$. Prove that the points $A, B, C, D$ are concyclic if and only if the points $A', B', C', D'$ are concyclic.

A computer outputs the values of the expression $ (n+1) . 2^n$ for $ n = 1, n = 2, n = 3$, etc. What is the largest number of consecutive values that are perfect squares?

Let a, b, c be positive integers. Prove that the inequality \[ (x-y)^a(x-z)^b(y-z)^c \ge 0 \] holds for all reals x, y, z if and only if a, b, c are even.