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## Balkan Mathematical Olympiad – 2016 Problems

Problem 1. Find all injective functions ${f: \mathbb R \rightarrow \mathbb R}$ such that for every real number ${x}$ and every positive integer ${n}$,

$\displaystyle \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016$

Problem 2. Let ${ABCD}$ be a cyclic quadrilateral with ${AB. The diagonals intersect at the point ${F}$ and lines ${AD}$ and ${BC}$ intersect at the point ${E}$. Let ${K}$ and ${L}$ be the orthogonal projections of ${F}$ onto lines ${AD}$ and ${BC}$ respectively, and let ${M}$, ${S}$ and ${T}$ be the midpoints of ${EF}$, ${CF}$ and ${DF}$ respectively. Prove that the second intersection point of the circumcircles of triangles ${MKT}$ and ${MLS}$ lies on the segment ${CD}$.

Problem 3. Find all monic polynomials ${f}$ with integer coefficients satisfying the following condition: there exists a positive integer ${N}$ such that ${p}$ divides ${2(f(p)!)+1}$ for every prime ${p>N}$ for which ${f(p)}$ is a positive integer.

Problem 4. The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of ${1201}$ colours so that no rectangle with perimeter ${100}$ contains two squares of the same colour. Show that no rectangle of size ${1\times1201}$ or ${1201\times1}$ contains two squares of the same colour.