What are the last two digits of the decimal representation of $21^{2006}$?

Consider all right triangles with integer sides such that the length of the hypotenuse and one of the two sides are consecutive. How many such triangles exist?

Let $\Gamma_A$, $\Gamma_B$, $\Gamma_C$ be circles such that $\Gamma_A$ is tangent to $\Gamma_B$ and $\Gamma_B$ is tangent to $\Gamma_C$. All three circles are tangent to lines $L$ and $M$, with $A$, $B$, $C$ being the tangency points of $M$ with $\Gamma_A$, $\Gamma_B$, $\Gamma_C$, respectively. Given that $12=r_A<r_B<r_C=75$, calculate: a) the length of $r_B$. b) the distance between point $A$ and the point of intersection of lines $L$ and $M$.

Consider all pairs of positive integers $(a,b)$, with $a<b$, such that $\sqrt{a} +\sqrt{b} = \sqrt{2,160}$ Determine all possible values of $a$.

Let $ABC$ be a triangle, and let $P$ be a point on side $BC$ such that $\frac{BP}{PC}=\frac{1}{2}$. If $\measuredangle ABC$ $=$ $45^{\circ}$ and $\measuredangle APC$ $=$ $60^{\circ}$, determine $\measuredangle ACB$ without trigonometry.