INMO 1994

Let $G$ be the centroid of the triangle $ABC$ in which the angle at $C$ is obtuse and $AD$ and $CF$ be the medians from $A$ and $C$ respectively onto the sides $BC$ and $AB$ . If the points $\ B,\ D, \ G$ and $\ F$ are concyclic, show that $\dfrac{AC}{BC} \geq \sqrt{2}$ . If further $P$ is a point on the line $BG$ extended such that $AGCP$ is a parallelogram, show that triangle $ABC$ and $GAP$ are similar.

If $x^5 - x ^3 + x = a,$ prove that $x^6 \geq 2a - 1$ .

In any set of $181$ square integers, prove that one can always find a subset of $19$ numbers, sum of whose elements is divisible by $19$ .

Find the number of nondegenerate triangles whose vertices lie in the set of points $(s,t)$ in the plane such that $0 \leq s \leq 4$ , $0 \leq t \leq 4$ , $s$ and $t$ are integers.

A circle passes through the vertex of a rectangle $ABCD$ and touches its sides $AB$ and $AD$ at $M$ and $N$ respectively. If the distance from $C$ to the line segment $MN$ is equal to $5$ units, find the area of rectangle $ABCD$ .