Let $G$ be the centroid of triangle $A B C$ and $M$ be the midpoint of $B C$. Let $X$ be on $A B$ and $Y$ on $A C$ such that the points $X, Y$, and $G$ are collinear and $X Y$ and $B C$ are parallel. Suppose that $X C$ and $G B$ intersect at $Q$ and $Y B$ and $G C$ intersect at $P$. Show that triangle $M P Q$ is similar to triangle $A B C$.
Given are two tangent circles and a point $P$ on their common tangent perpendicular to the lines joining their centres. Construct with ruler and compass all the circles that are tangent to these two circles and pass through the point $P$.
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?
Let $a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots+a_{n}=b_{1}+b_{2}+ \cdots+b_{n}$. Show that
At a party every woman dances with at least one man, and no man dances with every woman. Show that there are men $M$ and $M'$ and women $W$ and $W'$ such that $M$ dances with $W$, $M'$ dances with $W'$, but $M$ does not dance with $W'$, and $M'$ does not dance with $W$.
$P_0 = (1,0), P_1 = (1,1), P_2 = (0,1), P_3 = (0,0)$. $P_{n+4}$ is the midpoint of $P_n P_{n+1}$. $Q_n$ is the quadrilateral $P_n P_{n+1} P_{n+2} P_{n+3}$. $A_n$ is the interior of $Q_n$. Find $\bigcap_{n \ge 0} A_n$.
Given $k \gt 0$, the sequence $a_1, a_2, a_3, \dots$ is defined by its first two members and $a_{n+2} = a_{n+1} + \frac{k}{n} a_n$. For which $k$ can we write an $a_n$ as a polynomial in $n$? For which $k$ can we write $\frac{a_{n+1}}{a_n} = \frac{p(n)}{q(n)}$, where $p$ and $q$ are polynomials?
A castle has a number of halls and $n$ doors. Every door leads into another hall or outside. Every hall has at least two doors. A knight enters the castle. In any hall, he can choose any door for exit except the one he just used to enter that hall. Find a strategy allowing the knight to get outside after visiting no more than $2 n$ halls (a hall is counted each time it is entered).
An equilateral triangle is divided into $25$ congruent triangles enumerated with numbers from $1$ to $25$. Prove that one can find two triangles having a common side and with the difference of the numbers assigned to them greater than $3$.
There are 20 cats priced from $\$ 12$ to $\$ 15$ and 20 sacks priced from 10 cents to $\$ 1$ for sale (all prices are different). Prove that each of two boys, John and Peter, can buy a cat in a sack paying the same amount of money.
Find the smallest positive integer $n$ having the property: for any set of $n$ distinct integers $a_{1}, a_{2}, \ldots, a_{n}$ the product of all differences $a_{i}-a_{j}$, $i\lt j$, is divisible by $1991$.
Let's expand a little bit three circles, touching each other externally, so that three pairs of intersection points appear. Denote by $A_{1}, B_{1}, C_{1}$ the three so obtained "external" points and by $A_{2}, B_{2}, C_{2}$ the corresponding "internal" points. Prove the equality
$$
|A_{1} B_{2}| \cdot |B_{1} C_{2}| \cdot |C_{1} A_{2}| = |A_{1} C_{2}| \cdot |C_{1} B_{2}| \cdot |B_{1} A_{2}|.
$$
Let the coordinate planes have the reflection property. A beam falls onto one of them. How does the final direction of the beam after reflecting from all three coordinate planes depend on its initial direction?