MathDB

4

Part of 2009 Korea National Olympiad

Problems(2)

Double Counting on a simple Graph

Source: Korea National Olympiad 2009 Problem 4

9/9/2012
There are n(≥3)n ( \ge 3) students in a class. Some students are friends each other, and friendship is always mutual. There are s(≥1) s ( \ge 1 ) couples of two students who are friends, and t(≥1) t ( \ge 1 ) triples of three students who are each friends. For two students x,y x, y define d(x,y) d(x,y) be the number of students who are both friends with x x and y y . Prove that there exist three students u,v,w u, v, w who are each friends and satisfying d(u,v)+d(v,w)+d(w,u)≥9ts. d(u,v) + d(v,w) + d(w,u) \ge \frac{9t}{s} .
combinatorics proposedcombinatorics
Minimum value of a function at a interval [0,n+1]

Source: Korea National Olympiad 2009 Problem 8

9/9/2012
For a positive integer nn, define a function fn(x) f_n (x) at an interval [0,n+1] [ 0, n+1 ] as fn(x)=(∑i=1n∣x−i∣)2−∑i=1n(x−i)2. f_n (x) = ( \sum_{i=1} ^ {n} | x-i | )^2 - \sum_{i=1} ^{n} (x-i)^2 . Let an a_n be the minimum value of fn(x)f_n (x) . Find the value of ∑n=111(−1)n+1an. \sum_{n=1}^{11} (-1)^{n+1} a_n .
functionalgebra proposedalgebra