INMO 1989

Prove that the Polynomial $f(x) = x^{4} + 26x^{3} + 56x^{2} + 78x + 1989$ can't be expressed as a product $f(x) = p(x)q(x)$ , where $p(x)$ and $q(x)$ are both polynomial with integral coefficients and with degree at least $1$ .

Let $a,b,c$ and $d$ be any four real numbers, not all equal to zero. Prove that the roots of the polynomial $f(x) = x^{6} + ax^{3} + bx^{2} + cx + d$ can't all be real.

Let $A$ denote a subset of the set $\{ 1,11,21,31, \dots ,541,551 \}$ having the property that no two elements of $A$ add up to $552$ . Prove that $A$ can't have more than $28$ elements.

For positive integers $n$ , define $A(n)$ to be $\frac {(2n)!}{(n!)^{2}}$ . Determine the sets of positive integers $n$ for which

(a) $A(n)$ is an even number,

(b) $A(n)$ is a multiple of $4$ .

Triangle $ABC$ has incentre $I$ and the incircle touches $BC, CA$ at $D, E$ respectively. Let $BI$ meet $DE$ at $G$ . Show that $AG$ is perpendicular to $BG$ .