OPM 2020

In a triangle $[ABC]$ , $\angle C = 2\angle A$ . A point $D$ is marked on the side $[AC]$ such that $\angle ABD = \angle DBC$ . Knowing that $AB = 10$ and $CD = 3$ , what is the length of the side $[BC]$ ?

Given a subset of $\{1,2,...,n\}$ , we define its alternating sum in the following way: we order the elements of the subset in descending order and, starting with the largest, we alternately add and subtract the successive numbers. For example, the alternating sum of the set $\{1,3,4,6,8\}$ is $8-6+4-3+1 = 4$ . Determines the sum of the alternating sums of all subsets of $\{1,2,...,10\}$ with an odd number of elements.

Determine the fractions of a fraction of the form $\frac{1}{ab}$ where $a,b$ are prime natural numbers such that $0 < a < b \le 200$ and $a + b > 200$