MathDB

3

Part of 1985 Vietnam Team Selection Test

Problems(2)

Existence of triangle satisfies trigonometric conditions

Source: Vietnam TST 1985 Problem 3

2/5/2010
Does there exist a triangle ABC ABC satisfying the following two conditions: (a) sin⁡2A+sin⁡2B+sin⁡2C=cot⁡A+cot⁡B+cot⁡C{ \sin^2A + \sin^2B + \sin^2C = \cot A + \cot B + \cot C} (b) S≥a2−(b−c)2 S\ge a^2 - (b - c)^2 where S S is the area of the triangle ABC ABC.
trigonometrygeometrygeometry proposed
infinitely many points of discontinuity

Source: Vietnam TST 1985 Problem 6

2/5/2010
Suppose a function f:R→R f: \mathbb R\to \mathbb R satisfies f(f(x)) \equal{} \minus{} x for all x∈R x\in \mathbb R. Prove that f f has infinitely many points of discontinuity.
functionalgebrafunctional equationalgebra proposed