MathDB

3

Part of 1996 China National Olympiad

Problems(2)

Functional Equation involving cubes

Source: Chinese MO 1996

4/25/2010
Suppose that the function f:R→Rf:\mathbb{R}\to\mathbb{R} satisfies f(x3+y3)=(x+y)(f(x)2−f(x)f(y)+f(y)2)f(x^3 + y^3)=(x+y)(f(x)^2-f(x)f(y)+f(y)^2) for all x,y∈Rx,y\in\mathbb{R}. Prove that f(1996x)=1996f(x)f(1996x)=1996f(x) for all x∈Rx\in\mathbb{R}.
geometry3D geometryfunctionalgebra unsolvedalgebra
Minimum value of all the longest side

Source: CMO-1996-6

1/3/2006
In the triangle ABCABC, ∠C=90∘,∠A=30∘\angle{C}=90^{\circ},\angle {A}=30^{\circ} and BC=1BC=1. Find the minimum value of the longest side of all inscribed triangles (i.e. triangles with vertices on each of three sides) of the triangle ABCABC.
geometryCanada