Determine all pairs $(h, s)$ of positive integers with the following property: If one draws $h$ horizontal lines and another $s$ lines which satisfy
(i) they are not horizontal,
(ii) no two of them are parallel,
(iii) no three of the $h+s$ lines are concurrent,
then the number of regions formed by these $h+s$ lines is $1992$.
In a circle $C$ with centre $O$ and radius $r$, let $C_{1}, C_{2}$ be two circles with centres $O_{1}, O_{2}$ and radii $r_{1}, r_{2}$ respectively, so that each circle $C_{i}$ is internally tangent to $C$ at $A_{i}$ and so that $C_{1}, C_{2}$ are externally tangent to each other at $A$.
Prove that the three lines $O A, O_{1} A_{2}$, and $O_{2} A_{1}$ are concurrent.
Let $n$ be an integer such that $n\gt 3$. Suppose that we choose three numbers from the set $\{1,2, \ldots, n\}$. Using each of these three numbers only once and using addition, multiplication, and parenthesis, let us form all possible combinations.
a. Show that if we choose all three numbers greater than $n / 2$, then the values of these combinations are all distinct.
b. Let $p$ be a prime number such that $p \leq \sqrt{n}$. Show that the number of ways of choosing three numbers so that the smallest one is $p$ and the values of the combinations are not all distinct is precisely the number of positive divisors of $p-1$.
In a chess tournament each player plays every other player once. A player gets 1 point for a win, $\frac{1}{2}$ point for a draw and 0 for a loss. Both men and women played in the tournament and each player scored the same total of points against women as against men. Show that the total number of players must be a square.
Given positive real numbers $x_1, x_2, \dots, x_n$ find the polygon $A_0A_1 \dots A_n$ with $A_0A_1 = x_1$, $A_1A_2 = x_2$, $\dots$, $A_{n-1}A_n = x_n$ which has the greatest area.
Find all $n \gt 5$ for which it is possible to find a convex polyhedron with all $n$ faces congruent such that each face has another face parallel to it.
Let $d(n)$ be the number of positive divisors of $n$. Show that
$$
n\left(\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}\right) \le d(1) + d(2) + \dots + d(n) \le n\left(1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}\right)
$$
Find an infinite non-constant arithmetic progression of positive integers such that each term is neither a sum of two squares, nor a sum of two cubes (of positive integers).
Denote by $d(n)$ the number of all positive divisors of a positive integer $n$ (including $1$ and $n$). Prove that there are infinitely many $n$ such that $\frac{n}{d(n)}$ is an integer.
Find all fourth degree polynomials $p(x)$ such that the following four conditions are satisfied:
(i) $p(x) = p(-x)$ for all $x$.
(ii) $p(x) \geq 0$ for all $x$.
(iii) $p(0) = 1$.
(iv) $p(x)$ has exactly two local minimum points $x_1$ and $x_2$ such that $|x_1 - x_2| = 2$.
Noah has to fit 8 species of animals into 4 cages of the ark. He plans to put species in each cage. It turns out that, for each species, there are at most 3 other species with which it cannot share the accommodation. Prove that there is a way to assign the animals to their cages so that each species shares a cage with compatible species.