Let $a_{1}$ , $a_{2}$ , $a_{3}$ , $a_{4}$ be real numbers such that $a_{1}$ + $a_{2}$ + $a_{3}$ + $a_{4}$ = 0 and ${a^{2}}_{1}$ + ${a^{2}}_{2}$ + ${a^{2}}_{3}$ + ${a^{2}}_{4}$ = 1. Then, the smallest possible value of the expression $({a_{1} - a_{2})}^{2}$ + ${(a_{2} - a_{3})}^{2}$ + $({a_{3} - a_{4})}^{2}$ + ${{(a}_{4} - a_{1})}^{2}$ is:

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Detailed Solution

Let's find the smallest possible value of the expression $({a_{1} - a_{2})}^{2}$ + ${(a_{2} - a_{3})}^{2}$ + $({a_{3} - a_{4})}^{2}$ + ${{(a}_{4} - a_{1})}^{2}$

Given the conditions, $a_{1}$ + $a_{2}$ + $a_{3}$ + $a_{4}$ = 0 and ${a^{2}}_{1}$ + ${a^{2}}_{2}$ + ${a^{2}}_{3}$ + ${a^{2}}_{4}$ = 1.

First, we can simplify the expression:

$({a_{1} - a_{2})}^{2}$ + ${(a_{2} - a_{3})}^{2}$ + $({a_{3} - a_{4})}^{2}$ + ${{(a}_{4} - a_{1})}^{2}$

Expanding the squares:

$a_{1}^{2} + a_{2}^{2} - 2 a_{1} a_{2} + a_{2}^{2} + a_{3}^{2} - 2 a_{2} a_{3} + a_{3}^{2} + a_{4}^{2} - 2 a_{3} a_{4} + a_{4}^{2} + a_{1}^{2} - 2 a_{4} a_{1}$

Simplifying:

$2 (a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2}) - 2 (a_{1} a_{2} + a_{2} a_{3} + a_{3} a_{4} + a_{4} a_{1})$ ....(1)

Since we know $a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} = 1 and a_{1} + a_{2} + a_{3} + a_{4} = 0$ , we can rewrite the expression as:

$2 (1) - 2 (a_{1} a_{2} + a_{2} a_{3} + a_{3} a_{4} + a_{4} a_{1})$

Now, let's try to express $a_{1} a_{2} + a_{2} a_{3} + a_{3} a_{4} + a_{4} a_{1}$ in terms of the given conditions.

Squaring the sum $a_{1} + a_{2} + a_{3} + a_{4} = 0,$ we get:

${(a_{1} + a_{2} + a_{3} + a_{4})}^{2} = a_{1}^{2} + a_{2}^{2} + a_{3}^{2} + a_{4}^{2} + 2 (a_{1} a_{2} + a_{1} a_{3} + a_{1} a_{4} + a_{2} a_{3} + a_{2} a_{4} + a_{3} a_{4})$