MathDB

8

Part of 2006 All-Russian Olympiad

Problems(2)

Four zeros of iterated quadratic trinomial

Source: All-Russian Olympiad 2006 finals, problem 10.7 = 9.8

5/7/2006
Given a quadratic trinomial f(x)=x2+ax+bf\left(x\right)=x^2+ax+b. Assume that the equation f(f(x))=0f\left(f\left(x\right)\right)=0 has four different real solutions, and that the sum of two of these solutions is −1-1. Prove that b≤−14b\leq -\frac14.
quadraticsalgebra proposedalgebraquadratic equation
Tourists and t-shirts

Source: All-Russian Olympiad 2006 finals, problem 11.8

5/6/2006
At a tourist camp, each person has at least 5050 and at most 100100 friends among the other persons at the camp. Show that one can hand out a t-shirt to every person such that the t-shirts have (at most) 13311331 different colors, and any person has 2020 friends whose t-shirts all have pairwisely different colors.
ceiling functionfunctioncombinatorics proposedcombinatoricsProbabilistic Method