India National Olympiad 1997 Problems
Problem 1
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\[ AC^2 + CE \cdot CF = AB \cdot AE + AD \cdot AF . \]
Problem 2
Show that there do not exist positive integers $m$ and $n$ such that\[ \dfrac{m}{n} + \dfrac{n+1}{m} = 4 . \]
Problem 3
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 $abc + t = 0 .$
Problem 4
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 $\dfrac{14}{10} < p < \dfrac{15}{10}$.
Problem 5
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 $4$.
Problem 6
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\[ \dfrac{b}{a} < \alpha < \dfrac{3b}{2a}. \]
