The hyperbola $x^2 -y^2+ax + by =0$ and the circle $x^2+y^2 = a^2+b^2$ intersect in four points. Show that three of the four form an equilateral triangle

Let $D$ be any point in the interior of triangle $ABC$. Show that $|AD| + |DB| \leq |AC|+|CB|$.

Find the smallest positive integer with exactly 1000 positive divisors. Examples:

• 4 has 3 divisors $\{4,2,1\}$.
• 6 has 4 divisors $\{6,3,2,1\}$.
• 102 has 8 divisors $\{1,2,3,6,17,34,51,102\}$.