Subcontests
(6)CIIM 2013 Problem 6
Let (X,d) be a metric space with d:X×X→R≥0. Suppose that X is connected and compact. Prove that there exists an α∈R≥0 with the following property: for any integer n>0 and any x1,…,xn∈X, there exists x∈X such that the average of the distances from x1,…,xn to x is α i.e. nd(x,x1)+d(x,x2)+⋯+d(x,xn)=α. CIIM 2013 Problem 4
Let a1,b1,c1,a2,b2,c2 be positive real number and F,G:(0,∞)→(0,∞) be to differentiable and positive functions that satisfy the identities: Fx=1+a1x+b1y+c1G Gy=1+a2x+b2y+c2F.
Prove that if 0<x1≤x2 and 0<y2≤y1, then F(x1,x2)≤F(x2,y2) and G(x1,y1)≥G(x2,y2).
CIIM 2013 Problem 3
Given a set of boys and girls, we call a pair (A,B) amicable if A and B are friends. The friendship relation is symmetric. A set of people is affectionate if it satisfy the following conditions:i) The set has the same number of boys and girls.ii) For every four different people A,B,C,D if the pairs (A,B),(B,C),(C,D) and (D,A) are all amicable, then at least one of the pairs (A,C) and (B,D) is also amicable.
iii) At least 20131 of all boy-girl pairs are amicable.Let m be a positive integer. Prove that there exists an integer N(m) such that if a affectionate set has al least N(m) people, then there exists m boys that are pairwise friends or m girls that are pairwise friends.