Let $G$ be the centroid of triangle $ABC$ and $M$ the midpoint of $BC$. Let $X\in AB$, $Y\in AC$ such that $X,Y,G$ are collinear and $XY\parallel BC$. Suppose $XC$ and $GB$ intersect at $Q$, and $YB$ and $GC$ intersect at $P$. Show that triangle $MPQ$ is similar to triangle $ABC$.
Suppose there are $997$ points given in a plane. If every two points are joined by a line segment with its midpoint coloured in red, show that there are at least $1991$ red points in the plane. Can you find a special case with exactly $1991$ red points?
Schematic: only a few of the given points and one midpoint are shown.
Let $a_1,\ldots,a_n,b_1,\ldots,b_n$ be positive real numbers such that $a_1+\cdots+a_n=b_1+\cdots+b_n$. Show that
$$\frac{a_1^2}{a_1+b_1}+\frac{a_2^2}{a_2+b_2}+\cdots+\frac{a_n^2}{a_n+b_n}\ge\frac{a_1+a_2+\cdots+a_n}{2}.$$
During a break, $n$ schoolchildren sit in a circle around their teacher to play a game. Walking clockwise, the teacher chooses one child and gives that child a candy, skips the next child and gives a candy to the next, then skips $2$ children and gives a candy to the next, then skips $3$, and so on. Determine the values of $n$ for which eventually, perhaps after many rounds, every child receives at least one candy.
Given two tangent circles and a point $P$ on their common tangent perpendicular to the line joining their centres, construct with ruler and compass all circles tangent to these two circles and passing through $P$.
