Find the sum of all solutions.

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Let $5$ -tuples of positive integers $(x^{(1)}_{1},...,x^{(1)}_{5})$ , $(x^{(2)}_{1},...,x^{(2)}_{5})$ ,...., $(x^{(n)}_{1},..,x^{(n)}_{5})$ satisfy the following system of equations:

$x_{i}+x_{i+1}=x_{i+2}^{3}$ , $1≤i≤5$ $x_{6}=x_{1},x_{7}=x_{2}$ .

Find $\sum_{i=1}^{n}(x^{(i)}_{1}+...+x^{(i)}_{5})$

The answer is 10.

1 solution

Souryajit Roy
May 17, 2014

Let $x$ and $y$ be the largest and smallest of the numbers $x_{1},...,x_{5}$ .

Then, we get $x^{2}≤2x$ and $y^{2}≥2y$ .

Since $x$ and $y$ are both greater than o,we have $2≤y≤x≤2$ .Hence the equation has the unique solution $x_{1}=x_{2}=x_{3}=x_{4}=x_{5}=2$ .

Find the sum of all solutions.

The answer is 10.

1 solution

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