Two real numbers $x$ and $y$ satisfy $x+y=10$. The least possible value of $x^2+y^2$ is:

Write $y=10-x$, so $x^2+y^2=x^2+(10-x{)}^2=2x^2-20x+100=2(x-5{)}^2+50$. The square term is never negative, so the least value is 50, reached at $x=y=5$. (100 is $(x+y{)}^2$, which ignores the $-2xy$ correction; 25 is the greatest value of the product $xy$ at that same point; 20 is just twice the given sum.) Sample values: $x=4$ gives 16 + 36 = 52 and $x=5$ gives 25 + 25 = 50.