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Part of 1999 All-Russian Olympiad

Problems(3)

Inequality of sum of fractional parts

Source: All-Russian MO 1999

Prove that for all natural numbers

n

\sum_{k=1}^{n^2} \left\{ \sqrt{k} \right\} \le \frac{n^2-1}{2}.

\{x\}

denotes the fractional part of

x

inequalitiesinductionfloor functionalgebra unsolvedalgebranumber theory

Geometry Marathon-Olympiad level

Source: 0

Hi mathlinkers! I want to organise a Geometry marathon. This is to improve our skills in geometry. Whoever answers a pending problem can post a new problem. Let us start with easy ones and then go to complicated ones.
Different approaches are most welcome :).
Here is an easy one
1. Inradius of a triangle is equal to

1

.find the sides of the triangle and P.T one of its angle is

90

[color=#FF0000][moderator edit: topic capitalized. :)]

geometryinradiuscircumcircletrigonometryperimeterincenterratio

Line through three convex polygons

Source: All-Russian MO 1999

Three convex polygons are given on a plane. Prove that there is no line cutting all the polygons if and only if each of the polygons can be separated from the other two by a line.

geometryrotationcombinatorial geometrygeometry unsolved