SMO 2016 Open Round 2

Let $D$ be the point in the interior of a triangle $ABC$ such that $AB=ab$, $AC=ac$, $AD=ad$, $BC=bc$, $BD=bd$ and $CD=cd$. Prove that $\angle ABD+\angle ACD=60^\circ$.

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 $a_1,a_2,\ldots,a_n$ of $1,2,\ldots,n$ so that $a_1$, $a_1a_2$, $\ldots$, $a_1a_2\cdots a_n$ 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. The 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 $n$.