1975 USAMO Problems/Problem 3

Problem

If $P(x)$ denotes a polynomial of degree $n$ such that $P(k)=k/(k+1)$ for $k=0,1,2,\ldots,n$ , determine $P(n+1)$ .

Let $Q(x) = (x+1)P(x) - x$ . Clearly, $Q(x)$ has a degree of $n+1$ .

Thus, $k=0,1,2,\ldots,n$ are the roots of $Q(x)$ .

Since these are all $n+1$ of the roots, we can write $Q(x)$ as: $Q(x) = K(x)(x-1)(x-2) \cdots (x-n)$ where $K$ is a constant.

Plugging in $x = -1$ gives:

Finally, plugging in $x = n+1$ gives:

If $n$ is even, this simplifies to $P(n+1) = \dfrac{n}{n+2}$ . If $n$ is odd, this simplifies to $P(n+1) = 1$ .

1975 USAMO (Problems)
Preceded by Problem 2	1 • 2 • 3 • 4 • 5	Followed by Problem 4
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