BG-MO 1988

Find all real parameters $q$ for which there is a $p\in[0,1]$ such that the equation $$x^4+2px^3+(2p^2-p)x^2+(p-1)p^2x+q=0$$ has four real roots.

Let $n$ and $k$ be natural numbers and $p$ a prime number. Prove that if $k$ is the exact exponent of $p$ in $2^{2^n}+1$ (i.e. $p^k$ divides $2^{2^n}+1$ , but $p^{k+1}$ does not), then $k$ is also the exact exponent of $p$ in $2^{p-1}-1$ .

Let $M$ be an arbitrary interior point of a tetrahedron $ABCD$ , and let $S_A,S_B,S_C,S_D$ be the areas of the faces $BCD,ACD,ABD,ABC$ , respectively. Prove that $$S_A\cdot MA+S_B\cdot MB+S_C\cdot MC+S_D\cdot MD\ge9V,$$ where $V$ is the volume of $ABCD$ . When does equality hold?

Let $A,B,C$ be non-collinear points. For each point $D$ of the ray $AC$ , we denote by $E$ and $F$ the points of tangency of the incircle of $\triangle ABD$ with $AB$ and $AD$ , respectively. Prove that, as point $D$ moves along the ray $AC$ , the line $EF$ passes through a fixed point.

The points of space are painted in two colors. Prove that there is a tetrahedron such that all its vertices and its centroid are of the same color.

Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$ .