It is said that a positive integer is not GOOD, if there exists a permutation of the integers from 1 to n, $(a_1,a_2,...,a_n)$ such that $k + a_k$ is a perfect square for all $k$. For example $5$ is a GOOD number, since the permutation $(3,2,1,5,4)$ checks the condition: $1 + 3 = 2^2$, $2 + 2 = 2^2$, $3 + 1 = 2^2$; $4 + 5 = 3^2$ and $5 +4 = 3^2$. Find all GOOD numbers up to $12$.

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 $a + b > 200$