3

Part of 2005 Spain Mathematical Olympiad

Problems(2)

Spanish Mathematical Olympiad 2005, Problem 3

Source: 2005 Spanish Mathematical Olympiad

We will say that a triangle is multiplicative if the product of the heights of two of its sides is equal to the length of the third side. Given

ABC \dots XYZ

is a regular polygon with

n

sides of length

1

n-3

diagonals that go out from vertex

A

divide the triangle

ZAB

n-2

smaller triangles. Prove that each one of these triangles is multiplicative.

triangle with b+c=2a

Source: 2005 Spanish Mathematical Olympiad P6

In a triangle with sides

a, b, c

a

is the arithmetic mean of

b

c

. Prove that: a)

0^o \le A \le 60^o

. b) The height relative to side

a

is three times the inradius

r

. c) The distance from the circumcenter to side

a

R - r

R

is the circumradius.

geometryinradiuscircumradiusSides of a triangle