MathDB

1

Part of 1991 Greece National Olympiad

Problems(3)

a+b>2 if a+b<2ab - 1991 Greece MO Grade X p1

Source:

9/6/2024
Let a,ba, b be two reals such that a+b<2aba+b<2ab. Prove that a+b>2a+b>2
algebrainequalities
{f(f(x))=x+1, f: &amp;Zeta;-&gt;&amp;Zeta;

Source: 1991 Greece MO Grade XII p1

9/6/2024
Prove that there is no function f:Z→Zf: \mathbb{Z}\to\mathbb{Z} such that f(f(x))=x+1f(f(x))=x+1, for all x∈Zx\in\mathbb{Z}.
functionalgebrafunctional
P(x^3+1)=(P (x+1))^3

Source: 1991 Greece MO Grade XI p1

9/6/2024
Find all polynomials P(x)P(x) , such that P(x3+1)=(P(x+1))3P(x^3+1)=\left(P (x+1)\right)^3
algebrapolynomial