A positive rational number is written on the blackboard. Every minute Vasya replaces the number $ r $ written on the board with $ \sqrt {r + 1} $. Prove that someday he will get an irrational number.

For which natural $ n \geq 3 $ numbers from 1 to $ n $ can be arranged by a circle so that each number does not exceed $60$ % of the sum of its two neighbors?

Point $ O $ is the center of the circumscribed circle of an acute triangle $ Abc $. A certain circle passes through the points $ B $ and $ C $ and intersects sides $ AB $ and $ AC $ of a triangle. On its arc lying inside the triangle, points $ D $ and $ E $ are chosen so that the segments $ BD $ and $ CE $ pass through the point $ O $. Perpendicular $ DD_1 $ to $ AB $ side and perpendicular $ EE_1 $ to $ AC $ side intersect at $ M $. Prove that the points $ A $, $ M $ and $ O $ lie on the same straight line.

Given the disjoint finite sets of natural numbers $ A $ and $ B $, consisting of $ n $ and $ m $ elements, respectively. It is known that every natural number belonging to $ A $ or $ B $ satisfies at least one of the conditions $ k + 17 \in A $, $ k-31 \in B $. Prove that $ 17n = 31m $

50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine. Proposed by A. Khrabrov, incorrect translation from Hungarian

We call a positive integer good if the sum of the reciprocals of all its natural divisors are integers. Prove that if $ m $ is a good number, and $ p> m $ is a prime number, then $ pm $ is not good.

The incircle of triangle $ABC$ touches its sides $AB$ and $BC$ at points $P$ and $Q.$ The line $PQ$ meets the circumcircle of triangle $ABC$ at points $X$ and $Y.$ Find $\angle XBY$ if $\angle ABC = 90^\circ.$ Proposed by A. Smirnov

Zeroes and ones are arranged in all the squares of $n\times n$ table. All the squares of the left column are filled by ones, and the sum of numbers in every figure of the form [asy][asy]size(50); draw((2,1)--(0,1)--(0,2)--(2,2)--(2,0)--(1,0)--(1,2));[/asy][/asy] (consisting of a square and its neighbours from left and from below) is even. Prove that no two rows of the table are identical. Proposed by O. Vanyushina