MathDB

1

Part of 1988 China Team Selection Test

Problems(2)

A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) >= 0

Source: China TST 1988, problem 1

6/27/2005
Suppose real numbers A,B,CA,B,C such that for all real numbers x,y,zx,y,z the following inequality holds: A(x−y)(x−z)+B(y−z)(y−x)+C(z−x)(z−y)≥0.A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0. Find the necessary and sufficient condition A,B,CA,B,C must satisfy (expressed by means of an equality or an inequality).
inequalitiesquadraticsinequalities unsolved
f(x) = 3x + 2 iterated 100 times

Source: China TST 1988, problem 5

6/27/2005
Let f(x)=3x+2.f(x) = 3x + 2. Prove that there exists m∈Nm \in \mathbb{N} such that f100(m)f^{100}(m) is divisible by 19881988.
number theory unsolvednumber theory