OME 2013

Let $a,b,n$ positive integers with $a>b$ and $ab-1=n^2$ . Prove that $a-b \geq \sqrt{4n-3}$ and study the cases where the equality holds.

Find all the possible values of a positive integer $n$ for which the expression $S_n=x^n+y^n+z^n$ is constant for all real $x,y,z$ with $xyz=1$ and $x+y+z=0$ .

Let $k,n$ be positive integers with $n \geq k \geq 3$ . We consider $n+1$ points on the real plane with none three of them on the same line. We colour any segment between the points with one of $k$ possibilities. We say that an angle is a "bicolour angle" iff its vertex is one of the $n+1$ points and the two segments that define it are of different colours. Show that there is always a way to colour the segments that makes more than $n \Big\lfloor{\frac{n}{k}}\Big\rfloor^2 \frac{k(k-1)}{2}$ bicolour angles.

Are there infinitely many positive integers $n$ that can not be represented as $n = a^3+b^5+c^7+d^9+e^{11}$ , where $a,b,c,d,e$ are positive integers? Explain why.

Study if it there exist an strictly increasing sequence of integers $0=a_0<a_1<a_2<...$ satisfying the following conditions $i)$ Any natural number can be written as the sum of two terms of the sequence (not necessarily distinct). $ii)$ For any positive integer $n$ we have $a_n > \frac{n^2}{16}$

Let $ABCD$ a convex quadrilateral where: $|AB|+|CD|=\sqrt{2} |AC|$ and $|BC|+|DA|=\sqrt{2} |BD|$ What form does the quadrilateral have?