Find the number of ordered pairs of positive integers $(m,n)$, where $m,n\leq 10$, such that $m!+n!$ is a multiple of $10$.

$12$ distinct points are equally spaced around a circle. How many ways can Bryan choose $3$ points (not in any order) out of these $12$ points such that they form an acute triangle (Rotations of a set of points are considered distinct). [i]Proposed by Bryan Guo [/i]

Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

Let $ABC$ be an obtuse triangle with orthocenter $H$ and centroid $S$. Let $D$, $E$ and $F$ be the midpoints of segments $BC$, $AC$, $AB$, respectively. Show that the circumcircle of triangle $ABC$, the circumcircle of triangle $DEF$ and the circle with diameter $HS$ have two distinct points in common. [i](Josef Greilhuber)[/i]

Two mathematicians discuss two positive integers. One of them states that the square of the ratio between their product and their sum is exactly one more than this ratio. What is the smaller of these two numbers?

Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.

In a triangle $\triangle ABC$ with $\angle ABC < \angle BCA$, we define $K$ as the excenter with respect to $A$. The lines $AK$ and $BC$ intersect in a point $D$. Let $E$ be the circumcenter of $\triangle BKC$. Prove that \[\frac{1}{|KA|} = \frac{1}{|KD|} + \frac{1}{|KE|}.\]

Let $0 \le a \le b \le c \le 3$ Prove : $(a-b)(a^2-9)+(a-c)(b^2-9)+(b-c)(c^2-9) \le 36$

$\cos^210^{\circ}+\cos^250-\sin40^{\circ}\cdot\sin80^{\circ}$=________.

Consider a system of infinitely many spheres made of metal, with centers at points $(a, b, c) \in \mathbb Z^3$. We say that the system is stable if the temperature of each sphere equals the average temperature of the six closest spheres. Assuming that all spheres in a stable system have temperatures between $0^\circ C$ and $1^\circ C$, prove that all the spheres have the same temperature.

Let $S = \{(x, y) : x, y \in \{1, 2, 3, \dots, 2012\}\}$. For all points $(a, b)$, let $N(a, b) = \{(a - 1, b), (a + 1, b), (a, b - 1), (a, b + 1)\}$. Kathy constructs a set $T$ by adding $n$ distinct points from $S$ to $T$ at random. If the expected value of $\displaystyle \sum_{(a, b) \in T} | N(a, b) \cap T |$ is 4, then compute $n$. [i]Proposed by Lewis Chen[/i]

Given a graph with $99$ vertices and degrees in $\{81,82,\dots,90\}$, prove that there exist $10$ vertices of this graph with equal degrees and a common neighbour. [i]Proposed by Alireza Alipour[/i]

$T_1$ consists of a single branch from 0 to 1 in the complex plane. This branch then splits into two branches at the endpoint which each form a $135^\circ$ angle with the branch in $T_1$, and each branch has length $\frac{1}{\sqrt{2}}$. This process is repeated so that from each terminal branch in $T_n$, we form two more branches at angles $135^\circ$ with $\frac{1}{\sqrt{2}}$ the length. Let $L_n$ be the collection of the $2^{n-1}$ endpoints of the tree $T_n$. If multiple terminal branches end at the same point, then that point is counted multiple times in $L_n$. Shown below is $T_k$ for $k=1, 2, 3$ with dots at the points in $L_k$. Find the sum of $\ell^2$ over all points $\ell$ in $L_{10}$.

Let, $a\geq b\geq c >0$ be real numbers such that for all natural number $n$, there exist triangles of side lengths $a^{n} , b^{n} ,c^{n}$. Prove that the triangles are isosceles.

How many different ways (up to rotation) are there of labeling the faces of a cube with the numbers $1, 2,..., 6$?

Determine the smallest possible number $n> 1$ such that there exist positive integers $a_{1}, a_{2}, \ldots, a_{n}$ for which ${a_{1}}^{2}+\cdots +{a_{n}}^{2}\mid (a_{1}+\cdots +a_{n})^{2}-1$.

The real numbers $x, y$ and $z$ are such that $x^2 + y^2 + z^2 = 1$. a) Determine the smallest and the largest possible values of $xy + yz - xz$. b) Prove that there does not exist a triple $(x, y, z)$ of rational numbers, which attains any of the two values in a).

Let $ABC$ be a triangle with $AB=9$, $BC=10$, $CA=11$, and orthocenter $H$. Suppose point $D$ is placed on $\overline{BC}$ such that $AH=HD$. Compute $AD$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f(x)^2+f(2y+1)=x^2+f(y)+y+1\] for all reals $x$, $y$. [i](Proposed by Lim Yun Zhe)[/i]

In the base ten number system the number $526$ means $5 \cdot 10^2+2 \cdot 10 + 6$. In the Land of Mathesis, however, numbers are written in the base $r$. Jones purchases an automobile there for $440$ monetary units (abbreviated m.u). He gives the salesman a $1000$ m.u bill, and receives, in change, $340$ m.u. The base $r$ is: ${{ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8}\qquad\textbf{(E)}\ 12} $

Find all pairs of positive integers $x, y$ such that $x^3 - y^3 = 2005(x^2 - y^2)$.

Let $A$ be the set of all binary sequences of length $n$ and denote $o =(0, 0, \ldots , 0) \in A$. Define the addition on $A$ as $(a_1, \ldots , a_n)+(b_1, \ldots , b_n) =(c_1, \ldots , c_n)$, where $c_i = 0$ when $a_i = b_i$ and $c_i = 1$ otherwise. Suppose that $f\colon A \to A$ is a function such that $f(0) = 0$, and for each $a, b \in A$, the sequences $f(a)$ and $f(b)$ differ in exactly as many places as $a$ and $b$ do. Prove that if $a$ , $b$, $c \in A$ satisfy $a+ b + c = 0$, then $f(a)+ f(b) + f(c) = 0$.

$P$, $Q$, $R$ and $S$ are (distinct) points on a circle. $PS$ is a diameter and $QR$ is parallel to the diameter $PS$. $PR$ and $QS$ meet at $A$. Let $O$ be the centre of the circle and let $B$ be chosen so that the quadrilateral $POAB$ is a parallelogram. Prove that $BQ$ = $BP$ .

Let $a, b, c$ denote the lengths of the sides $BC,CA,AB$, respectively, of a triangle $ABC$. If $P$ is any point on the circumference of the circle inscribed in the triangle, show that $aPA^2+bPB^2+cPC^2$ is constant.

[b]1.[/b] Some proper partitions $P_1, \dots , P_n$ of a finite set $S$ (that is, partitions containing at least two parts) are called [i]independent[/i] if no matter how we choose one class from each partition, the intersection of the chosen classes is nonempty. Show that if the inequality $\frac{\left | S \right | }{2} < \left |P_1 \right | \dots \left |P_n \right |\qquad \quad (*)$ holds for some independent partitions, then $P_1, \dots , P_n$ is maximal in the sense that there is no partition $P$ such that $P,P_1, \dots , P_n$ are independent. On the other hand, show that inequality $(*)$ is not necessary for this maximality. ([b]C.20[/b]) [E. Gesztelyi]