MathDB

1

Part of 2008 South East Mathematical Olympiad

Problems(2)

set and subset

Source: China south east mathematical olympiad 2008 day1 problem 1

7/14/2013
Given a set S={1,2,3,…,3n},(n∈N∗)S=\{1,2,3,\ldots,3n\},(n\in N^*), let TT be a subset of SS, such that for any x,y,z∈Tx, y, z\in T (not necessarily distinct) we have x+y+z∉Tx+y+z\not \in T. Find the maximum number of elements TT can have.
combinatorics unsolvedcombinatoricsSet systemsAdditive combinatoricsRamsey Theory
find the maximal value

Source: China south east mathematical Olympiad 2008 day2 problem 5

7/15/2013
Let λ\lambda be a positive real number. Inequality ∣λxy+yz∣≤52|\lambda xy+yz|\le \dfrac{\sqrt5}{2} holds for arbitrary real numbers x,y,zx, y, z satisfying x2+y2+z2=1x^2+y^2+z^2=1. Find the maximal value of λ\lambda.
inequalitiesinequalities unsolved