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If r(x),s(x) are distinct elements of $x$ then 0<x<1

Source: USAMO 1987 Problem 3

July 24, 2011

algebrapolynomialinductionalgebra unsolved

Problem Statement

Construct a set

S

of polynomials inductively by the rules:
(i)

x\in S

; (ii) if

f(x)\in S

, then

xf(x)\in S

and

x+(1-x)f(x)\in S

.
Prove that there are no two distinct polynomials in

S

whose graphs intersect within the region

\{0 < x < 1\}