Given a triangle $ABC$, the internal bisectors through $A$ and $B$ meet the opposite sides in $D$ and $E$, respectively. Prove that \[DE \le (3 - 2\sqrt2)(AB + BC + CA)\] and determine the cases of equality.

Let $a_1, a_2,\cdots ,a_n$ be positive integers and let $a$ be an integer greater than $1$ and divisible by the product $a_1a_2\cdots a_n$. Prove that $a^{n+1} + a-1$ is not divisible by the product $(a + a_1 - 1)(a + a_2 - 1) \cdots (a + a_n - 1)$.

Given an integer $n\ge 2$, prove that \[\lfloor \sqrt n \rfloor + \lfloor \sqrt[3]n\rfloor + \cdots +\lfloor \sqrt[n]n\rfloor = \lfloor \log_2n\rfloor + \lfloor \log_3n\rfloor + \cdots +\lfloor \log_nn\rfloor\]. EditThanks to shivangjindal for pointing out the mistake (and sorry for the late edit)

Choose arbitrarily $n$ vertices of a regular $2n-$gon and colour them red. The remaining vertices are coloured blue. We arrange all red-red distances into a nondecreasing sequence and do the same with the blue-blue distances. Prove that the two sequences thus obtained are identical.

In triangle $ABC$, $\angle BAC = 94^{\circ},\ \angle ACB = 39^{\circ}$. Prove that \[ BC^2 = AC^2 + AC\cdot AB\].