INMO 1990

Given the equation $$x^4 + px^3 + qx^2 + rx + s = 0$$ has four real, positive roots, prove that (a) $pr - 16s \geq 0$ (b) $q^2 - 36s \geq 0$ with equality in each case holding if and only if the four roots are equal.

Determine all non-negative integral pairs $(x, y)$ for which $$(xy - 7)^2 = x^2 + y^2.$$

Let $f$ be a function defined on the set of non-negative integers and taking values in the same set. Given that

(a) $x - f(x) = 19\left[\frac{x}{19}\right] - 90\left[\frac{f(x)}{90}\right]$ for all non-negative integers $x$ ;

(b) $1900 < f(1990) < 2000$ ,

find the possible values that $f(1990)$ can take. (Notation : here $[z]$ refers to largest integer that is $\leq z$ , e.g. $[3.1415] = 3$ ).

Consider the collection of all three-element subsets drawn from the set $\{1,2,3,4,\dots,299,300\}$ . Determine the number of those subsets for which the sum of the elements is a multiple of 3.

Let $a$ , $b$ , $c$ denote the sides of a triangle. Show that the quantity $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}$$ must lie between the limits $3/2$ and 2. Can equality hold at either limits?

Triangle $ABC$ is scalene with angle $A$ having a measure greater than 90 degrees. Determine the set of points $D$ that lie on the extended line $BC$ , for which $$|AD|=\sqrt{|BD| \cdot |CD|}$$ where $|BD|$ refers to the (positive) distance between $B$ and $D$ .