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Prove that integration of this two functions from a to b is 1

Source: 2019 Jozsef Wildt International Math Competition

May 20, 2020

integrationcalculusfunction

Problem Statement

Let

f

g

h : [a, b] \to \mathbb{R}

, three integrable functions such that:

\int \limits_a^b fgdx=\int \limits_a^bghdx=\int \limits_a^bhfdx=\int \limits_a^bg^2dx\int \limits_a^bh^2dx=1

Then

\int \limits_a^bg^2dx=\int \limits_a^bh^2dx=1