Determine the largest possible value of the expression $ab+bc+ 2ac$ for non-negative real numbers $a, b, c$ whose sum is $1$.

Solve for real $x$ : $2^{2x} \cdot 2^{3\{x\}} = 11 \cdot 2^{5\{x\}} + 5 \cdot 2^{2[x]}$ (For a real number $x, [x]$ denotes the greatest integer less than or equal to x. For instance, $[2.5] = 2$, $[-3.1] = -4$, $[\pi ] = 3$. For a real number $x, \{x\}$ is defined as $x - [x]$.)

\[\begin{tabular}{ccccccccccccc} & & & & & & C & & & & & & \\ & & & & & C & O & C & & & & & \\ & & & & C & O & N & O & C & & & & \\ & & & C & O & N & T & N & O & C & & & \\ & & C & O & N & T & E & T & N & O & C & & \\ & C & O & N & T & E & S & E & T & N & O & C & \\ C & O & N & T & E & S & T & S & E & T & N & O & C \end{tabular}\] For how many paths consisting of a sequence of horizontal and/or vertical line segments, with each segment connecting a pair of adjacent letters in the diagram above, is the word CONTEST spelled out as the path is traversed from beginning to end? $\textbf{(A) }63\qquad\textbf{(B) }128\qquad\textbf{(C) }129\qquad\textbf{(D) }255\qquad \textbf{(E) }\text{none of these}$

Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

Fix positive integers $k,n$. A candy vending machine has many different colours of candy, where there are $2n$ candies of each colour. A couple of kids each buys from the vending machine $2$ candies of different colours. Given that for any $k+1$ kids there are two kids who have at least one colour of candy in common, find the maximum number of kids.

Given an acute triangle $ABC$, mark $3$ points $X, Y, Z$ in the interior of the triangle. Let $X_1, X_2, X_3$ be the projections of $X$ to $BC, CA, AB$ respectively, and define the points $Y_i, Z_i$ similarly for $i=1, 2, 3$. a) Suppose that $X_iY_i<X_iZ_i$ for all $i=1,2,3$, prove that $XY<XZ$. b) Prove that this is not neccesarily true, if triangle $ABC$ is allowed to be obtuse. [i]Proposed by Ivan Chan Kai Chin[/i]

Let $ n,\ m$ be positive integers and $ \alpha ,\ \beta$ be real numbers. Prove the following equations. (1) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)(x \minus{} \beta)\ dx \equal{} \minus{} \frac 16 (\beta \minus{} \alpha)^3$ (2) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)\ dx \equal{} \minus{} \frac {n!}{(n \plus{} 2)!}(\beta \minus{} \alpha)^{n \plus{} 2}$ (3) $ \int_{\alpha}^{\beta} (x \minus{} \alpha)^n(x \minus{} \beta)^mdx \equal{} ( \minus{} 1)^{m}\frac {n!m!}{(n \plus{} m \plus{} 1)!}(\beta \minus{} \alpha)^{n \plus{} m \plus{} 1}$

Let $ a,b,n $ be three natural numbers. Prove that there exists a natural number $ c $ satisfying: $$ \left( \sqrt{a} +\sqrt{b} \right)^n =\sqrt{ c+(a-b)^n} +\sqrt{c} $$ [i]Dan Popescu[/i]

Let $\mathcal{F}$ be the family of all nonempty finite subsets of $\mathbb{N} \cup \{0\}.$ Find all real numbers $a$ for which the series $$\sum_{A \in \mathcal{F}} \frac{1}{\sum_{k \in A}a^k}$$ is convergent.

Prove that for any $ \delta\in[0,1]$ and any $ \varepsilon>0$, there is an $ n\in\mathbb{N}$ such that $ \left |\frac{\phi (n)}{n}-\delta\right| <\varepsilon$.

From the interval $(2^{2n},2^{3n})$ are selected $2^{2n-1}+1$ odd numbers. Prove that there are two among the selected numbers, none of which divides the square of the other.

A frog jumps on the coordinate lattice, starting from the point $(1,1)$, according to the following rules: (i) From point $(a,b)$ the frog can jump to either $(2a,b)$ or $(a,2b)$; (ii) If $a>b$, the frog can also jump from $(a,b)$ to $(a-b,b)$, while for $a<b$ it can jump from $(a,b)$ to $(a,b-a)$. Can the frog get to the point: (a) $(24,40)$; (b) $(40,60)$; (c) $(24,60)$; (d) $(200,4)$?

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

Define the sequence $oa_n$ as follows: $oa_0=1, oa_n= oa_{n-1} \cdot cos\left( \dfrac{\pi}{2^{n+1}} \right)$. Find $\lim\limits_{n\rightarrow+\infty} oa_n$.

Let $f\in C^2[0,N]$ and $|f'(x)|<1$, $f''(x)>0$ for every $x\in [0, N]$. Let $0\leq m_0\ <m_1 < \cdots < m_k\leq N$ be integers such that $n_i=f(m_i)$ are also integers for $i=0,1,\ldots, k$. Denote $b_i=n_i-n_{i-1}$ and $a_i=m_i-m_{i-1}$ for $i=1,2,\ldots, k$. a) Prove that $$-1<\frac{b_1}{a_1}<\frac{b_2}{a_2}<\cdots < \frac{b_k}{a_k}<1$$ b) Prove that for every choice of $A>1$ there are no more than $N / A$ indices $j$ such that $a_j>A$. c) Prove that $k\leq 3N^{2/3}$ (i.e. there are no more than $3N^{2/3}$ integer points on the curve $y=f(x)$, $x\in [0,N]$).

2011 numbers are written on a board. For any three numbers, their sum is also among numbers written on the board. What is the smallest number of zeros among all 2011 numbers? (Author: I. Bogdanov)

We are given an infinite set of points in the plane such that any two of them have a distance of at most one. Prove that all the axes of symmetry of this set are concurrent, provided that there are at least two of them. [i]Proposed by David Anghel[/i]

The sides $a,b,c$ of triangle $ABC$ satisfy $a\le b\le c$. The circumradius and inradius of triangle $ABC$ are $R$ and $r$ respectively. Let $f=a+b-2R-2r$. Determine the sign of $f$ by the measure of angle $C$.

If an acute-angled triangle $ABC$ is given, construct an equilateral triangle $A'B'C'$ in space such that lines $AA',BB', CC'$ pass through a given point.

Each student chooses $1$ math problem and $1$ physics problem among $20$ math problems and $11$ physics problems. No same pair of problem is selected by two students. And at least one of the problems selected by any student is selected by at most one other student. At most how many students are there?

Find all integer values of $a$ such that the quadratic expression $(x+a)(x+1991) +1$ can be factored as a product $(x+b)(x+c)$ where $b,c$ are integers.

Let $n>1$ be a positive integer. Claire writes $n$ distinct positive real numbers $x_1, x_2, \dots, x_n$ in a row on a blackboard. In a $\textit{move},$ William can erase a number $x$ and replace it with either $\tfrac{1}{x}$ or $x+1$ at the same location. His goal is to perform a sequence of moves such that after he is done, the number are strictly increasing from left to right. [list] [*]Prove that there exists a positive constant $A,$ independent of $n,$ such that William can always reach his goal in at most $An \log n$ moves. [*]Prove that there exists a positive constant $B,$ independent of $n,$ such that Claire can choose the initial numbers such that William cannot attain his goal in less than $Bn \log n$ moves. [/list]

Compute the number of positive integers $n \leq 50$ such that there exist distinct positive integers $a,b$ satisfying \[ \frac{a}{b} +\frac{b}{a} = n \left(\frac{1}{a} + \frac{1}{b}\right). \]

Find the least positive integer $n$ for which there exists a set $\{s_1, s_2, \ldots , s_n\}$ consisting of $n$ distinct positive integers such that \[ \left( 1 - \frac{1}{s_1} \right) \left( 1 - \frac{1}{s_2} \right) \cdots \left( 1 - \frac{1}{s_n} \right) = \frac{51}{2010}.\] [i]Proposed by Daniel Brown, Canada[/i]

For two given triangles $A_1A_2A_3$ and $B_1B_2B_3$ with areas $\Delta_A$ and $\Delta_B$, respectively, $A_iA_k \ge B_iB_k, i, k = 1, 2, 3$. Prove that $\Delta_A \ge \Delta_B$ if the triangle $A_1A_2A_3$ is not obtuse-angled.