India National Olympiad 1989 Problems
Problem 1
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$.
Problem 2
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.
Problem 3
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.
Problem 4
Determine all $n \in \mathbb{N}$ for which
$n$ is not the square of any integer,
$\lfloor \sqrt{n}\rfloor ^3$ divides $n^2$.
Problem 5
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$.
Problem 6
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$.
Problem 7
Let $ A$ be one of the two points of intersection of two circles with centers $ X, Y$ respectively.The tangents at $ A$ to the two circles meet the circles again at $ B, C$. Let a point $ P$ be located so that $ PXAY$ is a parallelogram. Show that $ P$ is also the circumcenter of triangle $ ABC$.
