INMO 1997

Let $ABCD$ be a parallelogram. Suppose a line passing through $C$ and lying outside the parallelogram meets $AB$ and $AD$ produced at $E$ and $F$ respectively. Show that $$AC^2 + CE \cdot CF = AB \cdot AE + AD \cdot AF .$$

Show that there do not exist positive integers $m$ and $n$ such that $$\dfrac{m}{n} + \dfrac{n+1}{m} = 4 .$$

If $a,b,c$ are three real numbers and $$a + \dfrac{1}{b} = b + \dfrac{1}{c} = c + \dfrac{1}{a} = t$$ for some real number $t$ , prove that $abc + t = 0 .$

In a unit square one hundred segments are drawn from the centre to the sides dividing the square into one hundred parts (triangles and possibly quadruilaterals). If all parts have equal perimetr $p$ , show that $\dfrac{14}{10} < p < \dfrac{15}{10}$ .

Find the number of $4 \times 4$ array whose entries are from the set $\{ 0 , 1, 2, 3 \}$ and which are such that the sum of the numbers in each of the four rows and in each of the four columns is divisible by $4$ .

Suppose $a$ and $b$ are two positive real numbers such that the roots of the cubic equation $x^3 - ax + b = 0$ are all real. If $\alpha$ is a root of this cubic with minimal absolute value, prove that $$\dfrac{b}{a} < \alpha < \dfrac{3b}{2a}.$$