$A$, $B$, $C$, $D$ are four "consecutive" points on the circumference of a circle and $P$, $Q$, $R$, $S$ are points on the circumference which are respectively the midpoints of the $\operatorname{arcs}\ AB$, $BC$, $CD$, $DA$. Prove that $PR$ is perpendicular to $QS$.
For each real number $r$, $[r]$ denotes the largest integer less than or equal to $r$, e.g., $[6] = 6$, $[\pi] = 3$, $[-1.5] = -2$. Indicate on the $(x, y)$-plane the set of all points $(x, y)$ for which $[x]^2 + [y]^2 = 4$.
For a positive number such as $3.27$, $3$ is referred to as the integral part of the number and $.27$ as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.
A sequence of numbers $a_{1}, a_{2}, a_{3}, \ldots$ satisfies
(i) $a_{1}=\frac{1}{2}$,
(ii) $a_{1}+a_{2}+\cdots+a_{n}=n^{2} a_{n}$ \quad $(n \geq 1)$.
Determine the value of $a_{n}$ \quad $(n \geq 1)$.
i. $15$ chairs are equally placed around a circular table on which are name cards for $15$ guests. The guests fail to notice these cards until after they have sat down, and it turns out that no one is sitting in the correct seat. Prove that the table can be rotated so that at least two of the guests are simultaneously correctly seated.
ii. Give an example of an arrangement in which just one of the $15$ guests is correctly seated and for which no rotation correctly places more than one person.
A function $f(x)$ is periodic if there is a positive number $p$ such that $f(x+p)=f(x)$ for all $x$. For example, $\sin x$ is periodic with period $2 \pi$. Is the function $\sin \left(x^{2}\right)$ periodic? Prove your assertion.
Let $k$ be a positive integer. Find all polynomials
$$
P(x) = a_{0} + a_{1} x + \cdots + a_{n} x^{n}
$$
where the $a_{i}$ are real, which satisfy the equation
$$
P(P(x)) = [P(x)]^{k}
$$
A triangle $ABC$ has unit area. The first player chooses a point $X$ on side $AB$, then the second player chooses a point $Y$ on side $BC$, and finally the first player chooses a point $Z$ on side $CA$. The first player tries to arrange for the area of $XYZ$ to be as large as possible, the second player tries to arrange for the area to be as small as possible. What is the optimum strategy for the first player and what is the best he can do (assuming the second player plays optimally)?
$S$ is a horizontal strip in the plane. $n$ lines are drawn so that no three are collinear and every pair intersects within the strip. A path starts at the bottom border of the strip and consists of a sequence of segments from the $n$ lines. The path must change line at each intersection and must always move upwards. Show that:
(1) there are at least $n/2$ disjoint paths;
(2) there is a path of at least $n$ segments;
(3) there is a path involving not more than $n/2 + 1$ of the lines; and
(4) there is a path that involves segments from all $n$ lines.
Let $p(x)$ be a polynomial with integer coefficients. Let $f(n)$ be the sum of the (decimal) digits in the value $p(n)$. Show that $f(n)$ takes some value $m$ infinitely many times.
For what $n$ can we color the unit cubes in an $n \times n \times n$ cube red or green so that every red unit cube has just two red neighbouring cubes (sharing a face) and every green unit cube has just two green neighbouring cubes.
Given a convex hexagon, take the midpoint of each of the six diagonals joining vertices which are separated by a single vertex (so if the vertices are in order $A$, $B$, $C$, $D$, $E$, $F$, then the diagonals are $AC$, $BD$, $CE$, $DF$, $EA$, $FB$). Show that the midpoints form a convex hexagon with a quarter the area of the original.
The finite sequence $\{a_n\}$ has each member $0$, $1$ or $2$. A move involves replacing any two unequal members of the sequence by a single member different from either. A series of moves results in a single number. Prove that no series of moves can terminate in a (single) different number.
20 teams each play one game with every other team. Each game results in a win or loss (no draws). $k$ of the teams are European. A separate trophy is awarded for the best European team on the basis of the $k(k-1)/2$ games in which both teams are European. This trophy is won by a single team. The same team comes last in the overall competition (winning fewer games than any other team). What is the largest possible value of $k$? If draws are allowed and a team scores 2 for a win and 1 for a draw, what is the largest possible value of $k$?
(1) $O$ is the circumcenter of the triangle $ABC$. The triangle is rotated about $O$ to give a new triangle $A'B'C'$. The lines $AB$ and $A'B'$ intersect at $C''$, $BC$ and $B'C'$ intersect at $A''$, and $CA$ and $C'A'$ intersect at $B''$. Show that $A''B''C''$ is similar to $ABC$.
(2) $O$ is the center of the circle through $ABCD$. $ABCD$ is rotated about $O$ to give the quadrilateral $A'B'C'D'$. Prove that the intersection points of corresponding sides form a parallelogram.
Given real numbers $a_i$, $b_i$ and positive reals $c_i$, $d_i$, let $e_{ij} = (a_i + b_j) / (c_i + d_j)$. Let $M_i = \max_{0 \leq j \leq n} e_{ij}$, $m_j = \min_{1 \leq i \leq n} e_{ij}$. Show that we can find an $e_{ij}$ with $1 \leq i, j \leq n$ such that $e_{ij} = M_i = m_j$.