Difference between revisions of "2021 USAJMO Problems/Problem 6"

m (Solution)
m
 
(One intermediate revision by one other user not shown)
Line 2: Line 2:
 
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|after=Last Problem}}
+
 
 +
==See Also==
 +
{{USAJMO newbox|year=2021|num-b=5|after=Last Problem}}
  
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 18:06, 6 October 2023

Problem

Let $n \geq 4$ be an integer. Find all positive real solutions to the following system of $2n$ equations:\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*}

Solution

See Also

2021 USAJMO (ProblemsResources)
Preceded by
Problem 5
Followed by
Last Problem
1 2 3 4 5 6
All USAJMO Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png