MathDB

3

Part of 2002 France Team Selection Test

Problems(2)

a_{2i} is a permutation of {1,2,3...,n} iff a_1-a_2n=n

Source: French TST 2002

6/17/2011
Let nn be a positive integer and let (a1,a2,…,a2n)(a_1,a_2,\ldots ,a_{2n}) be a permutation of 1,2,…,2n1,2,\ldots ,2n such that the numbers ∣ai+1−ai∣|a_{i+1}-a_i| are pairwise distinct for i=1,…,2n−1i=1,\ldots ,2n-1. Prove that {a2,a4,…,a2n}={1,2,…,n}\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\} if and only if a1−a2n=na_1-a_{2n}=n.
absolute valuecombinatorics proposedcombinatorics
All sums a_i + a_j are distinct

Source: French TST 2002

6/17/2011
Let p≥3p\ge 3 be a prime number. Show that there exist pp positive integers a1,a2,…,apa_1,a_2,\ldots ,a_p not exceeding 2p22p^2 such that the p(p−1)2\frac{p(p-1)}{2} sums ai+aj (i<j)a_i+a_j\ (i<j) are all distinct.
number theory proposednumber theory