8

Part of 2008 Tuymaada Olympiad

Problems(2)

250 numbers are chosen in pos. integers not exceeding 501

Source: Tuymaada 2008, Junior League, Second Day, Problem 8.

250 numbers are chosen among positive integers not exceeding 501. Prove that for every integer

t

there are four chosen numbers

a_1

a_2

a_3

a_4

, such that a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \minus{} t is divisible by 23. Author: K. Kokhas

number theoryprime numbersnumber theory unsolved

Sum of the lengths of the three segments connecting midpoint

Source: Tuymaada 2008, Senior League, Second Day, Problem 8.

A convex hexagon is given. Let

s

be the sum of the lengths of the three segments connecting the midpoints of its opposite sides. Prove that there is a point in the hexagon such that the sum of its distances to the lines containing the sides of the hexagon does not exceed

s.

Author: N. Sedrakyan

analytic geometryinequalitiesgeometry unsolvedgeometry