The beam of a lighthouse on a small rock penetrates to a fixed distance $d$. As the beam rotates the extremity of the beam moves with velocity $v$. Prove that a ship with speed at most $v/8$ cannot reach the rock without being illuminated.
The $n^2$ numbers $\mathbf{x}_{ij}$ satisfy the $n^3$ equations: $\mathbf{x}_{ij} + \mathbf{x}_{jk} + \mathbf{x}_{ki} = 0$. Prove that we can find numbers $\mathbf{a}_1, \ldots, \mathbf{a}_n$ such that $\mathbf{x}_{ij} = \mathbf{a}_i - \mathbf{a}_j$.
Can 1965 points be arranged inside a square with side $15$ so that any rectangle of unit area placed inside the square with sides parallel to its sides must contain at least one of the points?
Given $n$ real numbers $\{a_1, a_2, \ldots, a_n\}$, prove that you can find $n$ integers $\{b_1, b_2, \ldots, b_n\}$, such that $|a_i - b_i| \lt 1$ and the sum of any subset of the original numbers differs from the sum of the corresponding $\{b_i\}$ by at most $(n + 1)/4$.
Two players have a $3 \times 3$ board. $9$ cards, each with a different number, are placed face up in front of the players. Each player in turn takes a card and places it on the board until all the cards have been played. The first player wins if the sum of the numbers in the first and third rows is greater than the sum in the first and third columns, loses if it is less, and draws if the sums are equal. Which player wins and what is the winning strategy?
Bus numbers have 6 digits, and leading zeros are allowed. A number is considered lucky if the sum of the first three digits equals the sum of the last three digits. Prove that the sum of all lucky numbers is divisible by 13.
A spy-plane circles point $A$ at a distance $10\mathrm{km}$ with speed $1000\mathrm{km/h}$. A missile is fired towards the plane from $A$ at the same speed and moves so that it is always on the line between $A$ and the plane. How long does it take to hit?
a. A committee has met 40 times, with 10 members at every meeting. No two people have met more than once at committee meetings. Prove that there are more than 60 people on the committee.
b. Prove that you cannot make more than 30 subcommittees of 5 members from a committee of 25 members with no two subcommittees having more than one common member.
A circle is circumscribed about the triangle $ABC$. $X$ is the midpoint of the arc $BC$ (on the opposite side of $BC$ to $A$), $Y$ is the midpoint of the arc $AC$, and $Z$ is the midpoint of the arc $AB$. $YZ$ meets $AB$ at $D$ and $YX$ meets $BC$ at $E$. Prove that $DE$ is parallel to $AC$ and that $DE$ passes through the center of the inscribed circle of $ABC$.
A tangent to the inscribed circle of a triangle drawn parallel to one of the sides meets the other two sides at $X$ and $Y$. What is the maximum length $XY$, if the triangle has perimeter $p$?
A tourist arrives in Moscow by train and wanders randomly through the streets on foot. After supper he decides to return to the station along sections of street that he has traversed an odd number of times. Prove that this is always possible. [In other words, given a path over a graph from $A$ to $B$, find a path from $B$ to $A$ consisting of edges that are used an odd number of times in the first path.]
An alien moves on the surface of a planet with speed not exceeding $u$. A spaceship searches for the alien with speed $v$. Prove the spaceship can always find the alien if $v \gt 10u$.
Given two relatively prime natural numbers $r$ and $s$, call an integer good if it can be represented as $mr + ns$ with $m, n$ non-negative integers and bad otherwise. Prove that we can find an integer $c$, such that just one of $k$, $c - k$ is good for any $k$. How many bad numbers are there?
A group of 100 people is formed to patrol the local streets. Every evening 3 people are on duty. Prove that you cannot arrange for every pair to meet just once on duty.