# IMO 1972 Q5 #

Problem: `f`

and `g`

are real-valued functions defined on the real line. For all `x`

and `y`

,
`f(x + y) + f(x - y) = 2f(x)g(y)`

. `f`

is not identically zero and `|f(x)| ≤ 1`

for all `x`

.
Prove that `|g(x)| ≤ 1`

for all `x`

.

This proof begins by introducing the supremum of `f`

, `k ≤ 1`

as well as `k' = k / ‖g y‖`

. We then
suppose that the conclusion does not hold (`hneg`

) and show that `k ≤ k'`

(by
`2 * (‖f x‖ * ‖g y‖) ≤ 2 * k`

obtained from the main hypothesis `hf1`

) and that `k' < k`

(obtained
from `hneg`

directly), finally raising a contradiction with `k' < k'`

.

(Authored by Stanislas Polu inspired by Ruben Van de Velde).

IMO 1972 Q5

Problem: `f`

and `g`

are real-valued functions defined on the real line. For all `x`

and `y`

,
`f(x + y) + f(x - y) = 2f(x)g(y)`

. `f`

is not identically zero and `|f(x)| ≤ 1`

for all `x`

.
Prove that `|g(x)| ≤ 1`

for all `x`

.

This is a more concise version of the proof proposed by Ruben Van de Velde.