3

Part of 2010 Poland - Second Round

Problems(2)

Polish 2nd stage 2010, 3rd problem (consecutive numbers)

Source:

Positive integer numbers

k

n

satisfy the inequality

k > n!

. Prove that there exist pairwisely different prime numbers

p_1, p_2, \ldots, p_n

which are divisors of the numbers

k+1, k+2, \ldots, k+n

respectively (i.e.

p_i|k+i

inequalitiesnumber theoryprime numbersnumber theory proposed

Polish 2nd stage 2010, 6th problem (arithmetic means)

Source:

n

-element set of real numbers is given, where

n \geq 6

. Prove that there exist at least

n-1

two-element subsets of this set, in which the arithmetic mean of elements is not less than the arithmetic mean of elements in the whole set.

ceiling functiongraph theorycombinatorics proposedcombinatorics