Olympiad problems

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Found problems: 127

2016 Czech-Polish-Slovak Junior Match, 1

Let $AB$ be a given segment and $M$ be its midpoint. We consider the set of right-angled triangles $ABC$ with hypotenuses $AB$. Denote by $D$ the foot of the altitude from $C$. Let $K$ and $L$ be feet of perpendiculars from $D$ to the legs $BC$ and $AC$, respectively. Determine the largest possible area of the quadrilateral $MKCL$. Czech Republic

2016 Saint Petersburg Mathematical Olympiad, 2

Tags: geometry , combinatorics , max , cube , 3-dimensional geometry , combinatorial geometry , 3d geometry

The rook, standing on the surface of the checkered cube, beats the cells, located in the same row as well as on the continuations of this series through one or even several edges. (The picture shows an example for a $4 \times 4 \times 4$ cube,visible cells that some beat the rook, shaded gray.) What is the largest number do not beat each other rooks can be placed on the surface of the cube $50 \times 50 \times 50$?