IMO Optimization Contest Problem: Find the minimum value of $xy$, given that $x^2+y^2+z^2=7$, $xy+xz+yz=4$, and $x, y$ and $z$ are real numbers.

IMO Optimization Contest Problem:

Find the minimum value of $xy$, given that $x^2+y^2+z^2=7$, $xy+xz+yz=4$, and $x, y$ and $z$ are real numbers.

My solution:

From the well-known identity

$(x+y+z)^2=x^2+y^2+z^2+2(xy+xz+yz)$

and the given values for $x^2+y^2+z^2=7$ and $xy+xz+yz=4$, we get:

$(x+y+z)^2=7+2(4)$