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

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a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \
 
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_{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} \
+
a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7}, \
&\vdots & &\vdots \
+
&\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}
 
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*}
 
\end{align*}

Revision as of 16:08, 15 April 2021

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

a1=1a2n+1a2,a2=a1+a3,a3=1a2+1a4,a4=a3+a5,a5=1a4+1a6,a6=a5+a7,a2n1=1a2n2+1a2n,a2n=a2n1+a1