Let $D$ be a point in the interior of $\triangle{ABC}$ such that $AB=ab$, $AC=ac$, $BC=bc$, $AD=ad$, $BD=bd$, $CD=cd$. Show that $\angle{ABD}+\angle{ACD}=60^{\circ}$. Source: 2016 Singapore Mathematical Olympiad (Open) Round 2, Problem 1

Let $a, b, c$ be real numbers such that $0 < a, b, c < 1/2$ and $a + b + c= 1$. Prove that for all real numbers $x,y,z$, $$abc(x + y + z)^2 \ge ayz( 1- 2a) + bxz( 1 - 2b) + cxy( 1 - 2c)$$. When does equality hold?

Let $n$ be a prime number. Show that there is a permutation $a_1,a_2,...,a_n$ of $1,2,...,n$ so that $a_1,a_1a_2,...,a_1a_2...a_n$ leave distinct remainders when divided by $n$

Let $b$ be a number with $-2 < b < 0$. Prove that there exists a positive integer $n$ such that all the coefficients of the polynomial $(x + 1)^n(x^2 + bx + 1)$ are positive.

A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimum value of $n$.