Given a triangle $ABC$ and a point $P_0$ on the side $AB$. Construct points $P_i$, $Q_i$, $R_i$ as follows: $Q_i$ is the foot of the perpendicular from $P_i$ to $BC$, $R_i$ is the foot of the perpendicular from $Q_i$ to $AC$ and $P_i$ is the foot of the perpendicular from $R_{i-1}$ to $AB$. Show that the points $P_i$ converge to a point $P$ on $AB$ and show how to construct $P$.
Box $A$ contains black balls and box $B$ contains white balls. Take a certain number of balls from $A$ and place them in $B$. Then take the same number of balls from $B$ and place them in $A$. Is the number of white balls in $A$ then greater, equal to, or less than the number of black balls in $B$?
A gambling student tosses a fair coin and scores one point for each head that turns up and two points for each tail. Prove that the probability of the student scoring exactly $n$ points is
$$
\frac{1}{3}\left[2+\left(-\frac{1}{2}\right)^{n}\right].
$$
The numbers from $1$ to $50$ are printed on cards. The cards are shuffled and then laid out face up in $5$ rows of $10$ cards each. The cards in each row are rearranged to make them increase from left to right. The cards in each column are then rearranged to make them increase from top to bottom. In the final arrangement, do the cards in the rows still increase from left to right?
A parallelepiped has the property that all cross sections which are parallel to any fixed face $F$, have the same perimeter as $F$. Determine whether or not any other polyhedron has this property.
$ABCD$ is a tetrahedron. Angles $ACB$ and $ADB$ are $90^\circ$. Let $k$ be the angle between the lines $AC$ and $BD$. Show that $\cos k \lt \dfrac{CD}{AB}$.
Some unit squares in an infinite sheet of squared paper are colored red so that every $2 \times 3$ and $3 \times 2$ rectangle contains exactly two red squares. How many red squares are there in a $9 \times 11$ rectangle?
A six digit (decimal) number has six different digits, none of them $0$, and is divisible by $37$. Show that you can obtain at least $23$ other numbers which are divisible by $37$ by permuting the digits.
There is a flu epidemic in elf city. The course of the disease is always the same. An elf is infected one day, he is sick the next, recovered and immune the third, recovered but not immune thereafter. Every day every elf who is not sick visits all his sick friends. If he is not immune he is sure to catch flu if he visits a sick elf. On day 1 no one is immune and one or more elves are infected from some external source. Thereafter there is no further external infection and the epidemic spreads as described above. Show that it is sure to die out (irrespective of the number of elves, the number of friends each has, and the number infected on day 1). Show that if one or more elves is immune on day 1, then it is possible for the epidemic to continue indefinitely.
35 containers of total weight 18 must be taken to a space station. One flight can take any collection of containers weighing 3 or less. It is possible to take any subset of 34 containers in 7 flights. Show that it must be possible to take all 35 containers in 7 flights.
All two digit numbers from $19$ to $80$ inclusive are written down one after the other as a single number $N = 192021\ldots 7980$. Is $N$ divisible by $1980$?
$S$ is a set of integers. Its smallest element is $1$ and its largest element is $100$. Every element of $S$ except $1$ is the sum of two distinct members of the set or double a member of the set. What is the smallest possible number of integers in $S$?
Given $1980$ vectors in the plane. The sum of every $1979$ vectors is a multiple of the other vector. Not all the vectors are multiples of each other. Show that the sum of all the vectors is zero.