India National Olympiad 1993 Problems
Problem 1
The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ intersect at $P$. Let $O$ be the circumcenter of triangle $APB$ and $H$ be the orthocenter of triangle $CPD$. Show that the points $H,P,O$ are collinear.
Problem 2
Let $p(x) = x^2 +ax +b$ be a quadratic polynomial with $a,b \in \mathbb{Z}$. Given any integer $n$ , show that there is an integer $M$ such that $p(n) p(n+1) = p(M)$.
Problem 3
If $a,b,c,d \in \mathbb{R}_{+}$ and $a+b +c +d =1$, show that\[ ab +bc +cd \leq \dfrac{1}{4}. \]
Problem 4
Let $ABC$ be a triangle in a plane $\pi$. Find the set of all points $P$ (distinct from $A,B,C$ ) in the plane $\pi$ such that the circumcircles of triangles $ABP$, $BCP$, $CAP$ have the same radii.
Problem 5
Show that there is a natural number $n$ such that $n!$ when written in decimal notation ends exactly in 1993 zeros.
Problem 6
Let $ABC$ be a triangle right-angled at $A$ and $S$ be its circumcircle. Let $S_1$ be the circle touching the lines $AB$ and $AC$, and the circle $S$ internally. Further, let $S_2$ be the circle touching the lines $AB$ and $AC$ and the circle $S$ externally. If $r_1, r_2$ be the radii of $S_1, S_2$ prove that $r_1 \cdot r_2 = 4 A[ABC]$.
Problem 7
Let $A = \{ 1,2, 3 , \ldots, 100 \}$ and $B$ be a subset of $A$ having $53$ elements. Show that $B$ has 2 distinct elements $x$ and $y$ whose sum is divisible by $11$.
Problem 8
Let $f$ be a bijective function from $A = \{ 1, 2, \ldots, n \}$ to itself. Show that there is a positive integer $M$ such that $f^{M}(i) = f(i)$ for each $i$ in $A$, where $f^{M}$ denotes the composition $f \circ f \circ \cdots \circ f$ $M$ times.
Problem 9
Show that there exists a convex hexagon in the plane such that
(i) all its interior angles are equal;
(ii) its sides are $1,2,3,4,5,6$ in some order.
