Difference between revisions of "2021 USAJMO Problems/Problem 6"
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Let <math>n \geq 4</math> be an integer. Find all positive real solutions to the following system of <math>2n</math> equations:<cmath>\begin{align*} a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\ a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\ a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7}, \\ &\vdots \\ a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1} \end{align*}</cmath> | Let <math>n \geq 4</math> be an integer. Find all positive real solutions to the following system of <math>2n</math> equations:<cmath>\begin{align*} a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\ a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\ a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7}, \\ &\vdots \\ a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1} \end{align*}</cmath> | ||
==Solution== | ==Solution== | ||
+ | {{USAJMO newbox|year=2021|num-b=4|num-a=Last problem!}} | ||
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+ | [[Category:Olympiad Number Theory Problems]] | ||
+ | {{MAA Notice}} |
Revision as of 09:59, 17 April 2021
Problem
Let be an integer. Find all positive real solutions to the following system of equations:
Solution
2021 USAJMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Problem Last problem! | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
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