Difference between revisions of "1984 USAMO Problems/Problem 5"
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== Solution == | == Solution == | ||
+ | By Lagrange Interpolation Formula <math>f(x) = 2\sum_{p=0}^{n}\left ( \prod_{0\leq r\neq3p\leq 3n}^{{}}\frac{x-r}{3p-r} \right )+ \sum_{p=1}^{n}\left ( \prod_{0\leq r\neq3p-2\leq 3n}^{{}} \frac{x-r}{3p-2-r}\right )</math> | ||
+ | and hence <math>f(3n+1) = 2\sum_{p=0}^{n}\left ( \prod_{0\leq r\neq3p\leq 3n}^{{}}\frac{3n+1-r}{3p-r} \right )+ \sum_{p=1}^{n}\left ( \prod_{0\leq r\neq3p-2\leq 3n}^{{}} \frac{3n+1-r}{3p-2-r}\right )</math> | ||
+ | |||
+ | after some calculations we get <math>f(3n+1) =\left ( \binom{3n+1}{0}- \binom{3n+1}{3}+\binom{3n+1}{6}- ... \right )\left ( 2.(-1)^{3n}-1 \right )+1</math> | ||
+ | |||
+ | Given <math>f(3n+1)= 730</math> so we have to find <math>n</math> such that <math>\left ( \binom{3n+1}{0}- \binom{3n+1}{3}+\binom{3n+1}{6}- ... \right )\left ( 2.(-1)^{3n}-1 \right )= 729</math> | ||
+ | |||
+ | Lemma: If <math>p</math> is even <math>\binom{p}{0}- \binom{p}{3}+ \binom{p}{6}- \cdots = \frac{2^{p+1}sin^{p}\left ( \frac{\pi}{3} \right )(i)^{p}\left ( cos\left ( \frac{p\pi}{3} \right ) \right )}{3}</math> | ||
+ | |||
+ | and if <math>p</math> is odd <math>\binom{p}{0}- \binom{p}{3}+ \binom{p}{6}- \cdots = \frac{-2^{p+1}sin^{p}\left ( \frac{\pi}{3} \right )(i)^{p+1}\left ( sin\left ( \frac{p\pi}{3} \right ) \right )}{3}</math> | ||
+ | |||
+ | <math>i</math> is <math>\sqrt{-1}</math> | ||
+ | Using above lemmas we do not get any solution when <math>n</math> is odd, but when <math>n</math> is even <math>3n+1=13 </math> satisfies the required condition, hence <math>n=4</math> | ||
{{solution}} | {{solution}} | ||
Revision as of 11:15, 22 April 2018
Problem
is a polynomial of degree such that
Determine .
Solution
By Lagrange Interpolation Formula
and hence
after some calculations we get
Given so we have to find such that
Lemma: If is even
and if is odd
is Using above lemmas we do not get any solution when is odd, but when is even satisfies the required condition, hence This problem needs a solution. If you have a solution for it, please help us out by adding it.
See Also
1984 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.