Difference between revisions of "1981 IMO Problems/Problem 6"
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{{IMO box|num-b=5|after=Last question|year=1981}} | {{IMO box|num-b=5|after=Last question|year=1981}} | ||
== Solution 2 == | == Solution 2 == | ||
+ | |||
+ | \documentclass{article} | ||
+ | \usepackage{multirow} | ||
+ | \begin{document} | ||
+ | \begin{tabular}{ |p{3cm}||p{3cm}|p{3cm}|p{3cm}| } | ||
+ | \hline | ||
+ | \multicolumn{4}{|c|}{f(x,y)} \ | ||
+ | \hline | ||
+ | x value & y=0 & y=1 & y=2 \ | ||
+ | \hline | ||
+ | & AF &AFG& 004\ | ||
+ | Aland Islands& AX & ALA &248\ | ||
+ | Albania &AL & ALB& 008\ | ||
+ | Algeria &DZ & DZA& 012\ | ||
+ | American Samoa& AS & ASM&016\ | ||
+ | Andorra& AD & AND &020\ | ||
+ | Angola& AO & AGO&024\ | ||
+ | \hline | ||
+ | \end{tabular} | ||
+ | \end{document} | ||
Revision as of 20:38, 8 April 2024
Problem
The function satisfies
(1)
(2)
(3)
for all non-negative integers . Determine .
Solution
We observe that and that , so by induction, . Similarly, and , yielding .
We continue with ; ; ; and ; .
It follows that when there are 1984 2s, Q.E.D.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
1981 IMO (Problems) • Resources | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Last question |
All IMO Problems and Solutions |
Solution 2
\documentclass{article} \usepackage{multirow} \begin{document} \begin{tabular}{ |p{3cm}||p{3cm}|p{3cm}|p{3cm}| }
\hline \multicolumn{4}{|c|}{f(x,y)} \\ \hline x value & y=0 & y=1 & y=2 \\ \hline & AF &AFG& 004\\ Aland Islands& AX & ALA &248\\ Albania &AL & ALB& 008\\ Algeria &DZ & DZA& 012\\ American Samoa& AS & ASM&016\\ Andorra& AD & AND &020\\ Angola& AO & AGO&024\\ \hline
\end{tabular} \end{document}