MathDB

3

Part of 1990 Greece National Olympiad

Problems(3)

equailteral of <CAD=<CBE=30^o - 1990 Greece MO Grade X p4

Source:

9/6/2024
In a triangle ABCABC with medians ADAD and BEBE , holds that ∠CAD=∠CBE=30o\angle CAD= \angle CBE=30^o. Prove that triangle ABCABC is equilateral.
geometryEquilateral
7 divides (1^n+2^n+3^n)

Source: 1990 Greece MO Grade XI p3

9/6/2024
For which nn, n∈N n \in \mathbb{N} is the number 1n+2n+3n1^n+2^n+3^n divisible by 77?
dividesdivisiblenumber theory
y^2f(x)(f(x)-2x)\le (1-xy)(1+xy)

Source: 1990 Greece MO Grade XII p3

9/6/2024
Find all functions f:R→Rf: \mathbb{R}\to\mathbb{R} that satisfy y2f(x)(f(x)−2x)≤(1−xy)(1+xy)y^2f(x)(f(x)-2x)\le (1-xy)(1+xy) for any x,y∈Rx,y \in\mathbb{R}.
functionalgebrainequalitiesfunctional