India National Olympiad 1990 Problems
Problem 1
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.
Problem 2
Determine all non-negative integral pairs $ (x, y)$ for which
\[ (xy – 7)^2 = x^2 + y^2.\]
Problem 3
Let $ f$ be a function defined on the set of non-negative integers and taking values in the same
set. Given that
(a) $ \displaystyle 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$).
Problem 4
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.
Problem 5
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?
Problem 6
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$.
Problem 7
Let $ ABC$ be an arbitrary acute angled triangle. For any point $ P$ lying within the triangle, let
$ D$, $ E$, $ F$ denote the feet of the perpendiculars from $ P$ onto the sides $ AB$, $ BC$, $ CA$ respectively.
Determine the set of all possible positions of the point $ P$ for which the triangle $ DEF$ is isosceles.
For which position of $ P$ will the triangle $ DEF$ become equilateral?
