Problems

You have an \(n\times m\) chocolate bar. You break the bar into two pieces along a line between its squares, then your friend and you take turns (your friend starts) choosing one of the pieces and breaking it again along a line between its squares. The player who cannot make a move loses. For which values of \(n\) and \(m\) do you win?

Four points \(A,B,C,D\) are chosen on the sides of a square of side length \(1\). The quadrilateral with vertices \(A,B,C,D\) has side lengths \(a,b,c,d\) as in the picture below. Show that \(2\leq a^2+b^2+c^2+d^2\leq 4\).

Let \(a, b, c\) be numbers such that \(a^2 + b^2 + c^2 = 1\). Show that \[-\frac12 \leq ab + bc + ac \leq 1.\]