Difference between revisions of "1975 USAMO Problems/Problem 3"

Revision as of 02:42, 21 May 2015

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 $P(n+1)$ .

Solution

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

Then, for $k=0,1,2,\ldots,n$ , $Q(k) = (k+1)P(k) - k = (k+1)\dfrac{k}{k+1} - k = 0$ .

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.

Thus, $(x+1)P(x) - x = K(x)(x-1)(x-2) \cdots (x-n)$

Plugging in $x = -1$ gives:

$(-1+1)P(-1) - (-1) = K(-1)(-1-1)(-1-2) \cdots (-1-n)$

$1 = K(-1)^{n+1}(1)(2) \cdots (n+1)$

$K = (-1)^{n+1}\dfrac{1}{(n+1)!}$

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

$(n+1+1)P(n+1) - (n+1) =$ $(-1)^{n+1}\dfrac{1}{(n+1)!}(n+1)(n+1-1)(n+1-2) \cdots (n+1-n)$

$(n+2)P(n+1) = (-1)^{n+1}\dfrac{1}{(n+1)!}\cdot(n+1)! + (n+1)$

$(n+2)P(n+1) = (-1)^{n+1} + (n+1)$

$P(n+1) = \dfrac{(-1)^{n+1} + (n+1)}{n+2}$

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

Solution 2

It is fairly natural to use Lagrange's Interpolation Formula on this problem:

$ $\begin{aligned} P (n + 1) & = \sum_{k = 0}^{n} \frac{k}{k + 1} \prod_{j not k} \frac{n + 1 - j}{k - j} \\ = \sum_{k = 0}^{n} \frac{k}{k + 1} \cdot \frac{\frac{(n + 1)!}{n + 1 - k}}{k (k - 1) (k - 2) \dots 1 \cdot (- 1) (- 2) \dots (k - n)} \\ = \sum_{k = 0}^{n} \frac{k}{k + 1} (- 1)^{n - k} \cdot \frac{(n + 1)!}{k! (n + 1 - k)!} \\ = \sum_{k = 0}^{n} (- 1)^{n - k} (\binom{n + 1}{k}) - \sum_{k = 0}^{n} \frac{(n + 1)! (- 1)^{n - k}}{(k + 1)! (n + 1 - k)!} \\ = - (\sum_{k = 0}^{n + 1} (- 1)^{n + 1 - k} (\binom{n + 1}{k}) - 1) + \frac{1}{n + 2} \cdot \sum_{k = 0}^{n} (- 1)^{n + 1 - k} (\binom{n + 2}{k + 1}) \\ = 1 + \frac{1}{n + 2} (\sum_{k = - 1}^{n + 1} (- 1)^{n + 2 - (k + 1)} (\binom{n + 2}{k + 1}) - (- 1)^{n + 2} - 1) \\ = 1 - \frac{(- 1)^{n} + 1}{n + 2} \end{aligned}$ $ (Error compiling LaTeX. Unknown error_msg)

through usage of the Binomial Theorem.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

@@ Line 1: / Line 1: @@
 ==Problem==
-If <math>P(x)</math> denotes a polynomial of degree <math>n</math> such that <math>P(k)=k/(k+1)</math> for <math>k=0,1,2,\ldots,n</math>, determine <math>P(n+1)</math>.
+If <math>P(x)</math> denotes a polynomial of degree <math>n</math> such that <cmath>P(k)=k/(k+1)</cmath> for <math>k=0,1,2,\ldots,n</math>, determine <math>P(n+1)</math>.
 ==Solution==

1975 USAMO (Problems • Resources)
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Difference between revisions of "1975 USAMO Problems/Problem 3"

Revision as of 02:42, 21 May 2015

Contents

Problem

Solution

Solution 2

See Also