Subcontests
(6)Sums of pairs in a sequence
Let n>1 be an integer. Find, with proof, all sequences x1,x2,…,xn−1 of positive integers with the following three properties:
(a). x1<x2<⋯<xn−1 ;
(b). xi+xn−i=2n for all i=1,2,…,n−1;
(c). given any two indices i and j (not necessarily distinct) for which xi+xj<2n, there is an index k such that xi+xj=xk. Intersecting Permutations
Two permutations a1,a2,…,a2010 and b1,b2,…,b2010 of the numbers 1,2,…,2010 are said to intersect if ak=bk for some value of k in the range 1≤k≤2010. Show that there exist 1006 permutations of the numbers 1,2,…,2010 such that any other such permutation is guaranteed to intersect at least one of these 1006 permutations. Permutations Part 1: 2010 USAJMO #1
A permutation of the set of positive integers [n]={1,2,...,n} is a sequence (a1,a2,…,an) such that each element of [n] appears precisely one time as a term of the sequence. For example, (3,5,1,2,4) is a permutation of [5]. Let P(n) be the number of permutations of [n] for which kak is a perfect square for all 1≤k≤n. Find with proof the smallest n such that P(n) is a multiple of 2010.