MathDB

trinomials f,g,h are sides of a triangle, but f(x)-1,g(x) -1, h(x) -1 are not

Source: Tuymaada Junior 2006 p5

May 12, 2019

algebraquadratic trinomialtrinomialquadraticsconstantSides of a triangle

Problem Statement

The quadratic trinomials

f

g

and

h

are such that for every real

x

the numbers

f (x)

g (x)

and

h (x)

are the lengths of the sides of some triangles, and the numbers

f (x) -1

g (x) -1

and

h (x) -1

are not the lengths of the sides of the triangle. Prove that at least of the polynomials

f + g-h

f + h-g

g + h-f

is constant.