## Romanian TST II 2011 Problem 4

Show that

(a) There are infinitely many positive integers such that there exists a square equal to the sum of the squares of consecutive positive integers (for instance, and are such, since , and ).

(b) If is a positive integer which is not a perfect square, and if is an integer number such that is a perfect square, then there are infinitely many positive integers such that is a perfect square.

Romanian TST 2011

## Romanian TST II Problem 3

Given a positive integer , determine the maximum number of edges a simple graph of edges vertices may have, in order that it won’t contain any cycles of even length.

Romanian TST 2011

## Romanian TST II 2011 Problem 2

Let be a triangle. The incircle of the triangle touches the side at the point (all indices are considered modulo ). Let be the foot of the perpendicular dropped from the point onto the line . Show that the lines are concurrent at a point situated on the Euler line of the triangle .

Romanian TST 2011

## Romanian TST II 2011 Problem 1

A square of sidelength is contained in the unit square whose centre is not interior to the former. Show that .

Romanian TST 2011

## Romanian TST 2011 Problem 4

Suppose and denote the permutations of the set . For denote by the number of cycles in the disjoint cycle decomposition of the permutation . Calculate the sum .

## Romanian TST 2011 Problem 2

Denote .

Prove that:

a) For any positive integer there exists an arithmetic progression of length in .

b) Prove that does not contain a infinite length arithmetic progression.