1987 USAMO Problems/Problem 3

Problem

$X$ is the smallest set of polynomials $p(x)$ such that:

1. $p(x) = x$ belongs to $X$ .

2. If $r(x)$ belongs to $X$ , then $x\cdot r(x)$ and $(x + (1 - x) \cdot r(x) )$ both belong to $X$ .

Show that if $r(x)$ and $s(x)$ are distinct elements of $X$ , then $r(x) \neq s(x)$ for any $0 < x < 1$ .

Solution

Let $s(x)$ be an arbitrary polynomial in $X.$ Then $0<s(x)<1$ when $0<x<1.$ Define $X_1=\{s(x)\in X:s(x)=x\cdot s_1(x)$ for some $s_1(x)\in X\},$ and $X_2=\{t(x)\in X: t(x)=x+(1-x)t_1(x)$ for some $t_1(x)\in X\}.$

If $s(x) \in X_1$ and $t(x)\in X_2,$ we have $s(x) <x<t(x)$ for all $x$ with $0<x<1.$ Therefore $s(x)\ne t(x)$ for any $0<x<1.$

For any $s(x), t(x) \in X_1$ , Let $s(x)=x\cdot s_1(x)$ and $t(x)=x\cdot t_1(x)$ for $s_1(x), t_1(x) \in X.$ If $s_1(x) \ne t_1(x)$ for $0<x<1,$ then $s(x)-t(x)=x(s_1(x)-t_1(x))\ne 0$ for $0<x<1.$

Similarly, for any $s(x), t(x) \in X_2$ , Let $s(x)=x+(1-x) s_1(x)$ and $t(x)=x+(1-x) t_1(x)$ for $s_1(x), t_1(x) \in X.$ If $s_1(x) \ne t_1(x)$ for $0<x<1,$ then $s(x)-t(x)=(1-x)(s_1(x)-t_1(x))\ne 0$ for $0<x<1.$

The proof is done by an induction.

J.Z.

1987 USAMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5
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1987 USAMO Problems/Problem 3

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