1. Let $d$ be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than $d$ degrees. What is the minimum possible value for $d$ ?
Answer:
120

The sum of the internal angles of a quadrilateral triangle is $360^{\circ}$. To find the minimum $d$, we note the limiting case where three of the angles have measure $d$ and the remaining angle has measure approaching zero. Hence, $d \geq 360^{\circ} / 3=120$. It is not difficult to see that for any $0<\alpha<120$, a quadrilateral of which three angles have measure $\alpha$ degrees and fourth angle has measure $(360-3 \alpha)$ degrees can be constructed.