Question

Find three real numbers whose sum is 22 and whose sum of squares is as small as possible

Answer 1

If the 3 numbers are the same we have 22/3 as the numbers. Sum of squares is 3×484/9=484/3=161⅓.
If we now take 7, 8 and 7 as the three numbers, sum of squares is 98+64=162 which is bigger than 161⅓.
If we take 6, 7 and 9 the sum of the squares is 36+49+81=166, bigger again.
So it would appear that the minimum sum of squares is when each number is 22/3 or 7⅓.