Home > Algebra, Combinatorics, Geometry, Olympiad, Uncategorized > Balkan Mathematical Olympiad – 2016 Problems

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<CD}. 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.

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