Two touching balls with radii $a$ and $b$ are enclosed in a cylindrical tin of diameter $d$ . Both balls hit the top surface and the shell of the cylinder. The largest ball also hits the bottom surface. Show that $\sqrt{d} =\sqrt{a} +\sqrt{b}$

Calculate the sum of all unsimplified fractions whose numerator and denominator are positive divisors of $1000$.

In triangle $\vartriangle ABC$ holds $\angle A= 40^o$ and $\angle B = 20^o$ . The point $P$ lies on the line $AC$ such that $C$ is between $A$ and $P$ and $| CP | = | AB | - | BC |$. Calculate the $\angle CBP$.

The Knights of the Round Table are gathering. Around the table are $34 $ chairs, numbered from 1 to $34$. When everyone has sat down, it turns out that between every two knights there is a maximum of $r$ places, which can be either empty or occupied by another knight. (a) For each $r \le 15$, determine the maximum number of knights present. (b) Determine for each $r \le 15$ how many sets of occupied seats there are that match meet the given and where the maximum number of knights is present.