A table consists of an $m\times m$ array of unit squares. Some square centres initially contain one ant each. From time $0$, every ant moves at speed $1$ parallel to a side of a square. When ants meet:
(i) If exactly two meet head-on, both turn clockwise through $90^\circ$ and continue at speed $1$.
(ii) If two meet at right angles, or more than two meet, they keep their velocities and directions.
An ant reaching the edge falls off and never returns. The time when the last ant falls is called their “doomsday”. Over all possible initial placements, find the latest possible doomsday, or prove that doomsday need not occur.
We have a cylinder with a radius of $1$ cm and a height of $4$ cm. While the point $P$ lies on the bottom circle of the cylinder, the point $Q$ lies on the top circle right above $P$. A string connecting the point $P$ and $Q$ makes a complete turn round the cylinder (see the picture). How long is the shortest such string in cm?
Which two points lie on the graph of a linear function $y = bx + 1$ where $b$ is some non-zero real number?
(A) $(0,1)$ and $(\frac{1}{b},0)$
(B) $(0,b)$ and $(-\frac{1}{b},0)$
(C) $(0,1)$ and $(b,0)$
(D) $(0,1)$ and $(-\frac{1}{b},0)$
(E) $(0,-\frac{1}{b})$ and $(1,0)$
Each positive integer is coloured red or blue. A function $f$ from the set of positive integers to itself has the following two properties:
(a) if $x \le y$, then $f(x) \le f(y)$; and
(b) if $x, y$ and $z$ are (not necessarily distinct) positive integers of the same colour and $x + y = z$, then $f(x) + f(y) = f(z)$.
Prove that there exists a positive number $a$ such that $f(x) \le a x$ for all positive integers $x$.
The points $A_1,A_2,\dots,A_{2n+1}$ are the vertices of a regular $(2n+1)$-gon. Find the number of triples $A_i,A_j,A_k$ for which triangle $A_iA_jA_k$ is obtuse.
Let $ABC$ be a right-angled triangle with the right angle at $C$. Let the points $K$, $L$ and $M$ lie on the sides $CA$, $AB$ and $BC$, respectively, so that the angle $\angle MLK$ is a right angle and $|KC| = |KL|$. Prove that the bisectors of the angles $\angle AKL$ and $\angle LMB$ are parallel.
Let a point $E$ lie on the side $CD$ of a square $ABCD$. Let a point $F$ lie on the line $AB$, but not on the line segment $AB$, and let $|BF| = |DE|$. Prove that the lines $AC$ and $EF$ are perpendicular.
Let $f: \mathbb{Q}^{+} \to \mathbb{R}^{+}$ be a function that satisfies
$$
f(x + y) - f(x - y) = 4\sqrt{f(x)f(y)}
$$
for all $0 \lt y \lt x$.
(a) Prove that $f(2x) = 4f(x)$ for all $x \in \mathbb{Q}^{+}$.
(b) Find all such functions.
Let $S=\{1,2,\dots,2012\}$. How many functions $f:S\to S$ satisfy both conditions?
(1) $f$ is a bijection.
(2) For every $1\le a\le2012$, $f(a)+f^{-1}(a)=2013$, where $f^{-1}$ is the inverse of $f$.
Let $ABC$ be a triangle. Let $D$ denote the intersection point of the bisector of the angle $\angle BAC$ and the side $BC$, and let $E$ denote the intersection point of the bisector of the angle $\angle CBA$ and the side $AC$. Suppose $|CD| = |CE|$. Prove that the triangle $ABC$ is isosceles.
Let $ABC$ be an acute non-isosceles triangle with circumcentre $O$ and incentre $I$. Its incircle touches $BC,CA,AB$ at $D,E,F$, respectively. Let $P=AI\cap OD$, $Q=BI\cap OE$, $R=CI\cap OF$, and let $M$ be the circumcentre of $PQR$. Prove that $I,M,O$ are collinear.
Let $O$ be the circumcircle of triangle $ABC$. The bisector of $\angle A$ meets $O$ again at $P$ and meets $BC$ at $D$. The perpendicular from $D$ to $AB$ meets $AB$ at $E$ and meets $O$ at $Q$, chosen so that $E$ lies on segment $DQ$. Let $F=PQ\cap BC$ and $G=AF\cap CQ$. Prove that $EG\parallel FC$.
Prove that there is no injective function $f: \mathbb{R} \to \mathbb{R}$ such that
$$
f(f(x) + y) = f(x + y) + f(2012) \quad \text{for all } x, y \in \mathbb{R}.
$$
