Subcontests
(6)There exists a permutation - Poland Third Round 2003
Let n be an even positive integer. Show that there exists a permutation (x1,x2,…,xn) of the set {1,2,…,n}, such that for each i∈{1,2,…,n},xi+1 is one of the numbers 2xi,2xi−1,2xi−n,2xi−n−1, where xn+1=x1. x^2 + y^2 + z^2 is divisible by x + y + z - Poland 2003
A prime number p and integers x,y,z with 0<x<y<z<p are given. Show that if the numbers x3,y3,z3 give the same remainder when divided by p, then x2+y2+z2 is divisible by x+y+z. Inequality with strictly increasing sequences - Poland 2003
Let 0<a<1 be a real number. Prove that for all finite, strictly increasing sequences k1,k2,…,kn of non-negative integers we have the inequality
(i=1∑naki)2<1−a1+ai=1∑na2ki.