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 ==https://beastacademy.com/ |
− | + | \begin{center} | |
− | + | \begin{tabular}{||c c c c||} | |
− | \begin{ | ||
− | \begin{tabular}{ | ||
\hline | \hline | ||
− | \ | + | Col1 & Col2 & Col2 & Col3 \\ [0.5ex] |
+ | \hline\hline | ||
+ | 1 & 6 & 87837 & 787 \ | ||
\hline | \hline | ||
− | + | 2 & 7 & 78 & 5415 \ | |
\hline | \hline | ||
− | + | 3 & 545 & 778 & 7507 \ | |
− | + | \hline | |
− | + | 4 & 545 & 18744 & 7560 \ | |
− | + | \hline | |
− | + | 5 & 88 & 788 & 6344 \ [1ex] | |
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\hline | \hline | ||
\end{tabular} | \end{tabular} | ||
− | \end{ | + | \end{center} |
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Revision as of 20:40, 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 ==https://beastacademy.com/
\begin{center} \begin{tabular}{||c c c c||}
\hline Col1 & Col2 & Col2 & Col3 \\ [0.5ex] \hline\hline 1 & 6 & 87837 & 787 \\ \hline 2 & 7 & 78 & 5415 \\ \hline 3 & 545 & 778 & 7507 \\ \hline 4 & 545 & 18744 & 7560 \\ \hline 5 & 88 & 788 & 6344 \\ [1ex] \hline
\end{tabular} \end{center}