Suppose that in a unit sphere in Euclidean space, there are $2m$ points $x_1, x_2, ..., x_{2m}.$ Prove that it's possible to partition them into two sets of $m$ points in such a way that the centers of mass of these sets are at a distance of at most $\frac{2}{\sqrt{m}}$ from one another.

The incircle of the triangle $ABC$ touches the sides $BC, CA$ and $AB{}$ at $D,E$ and $F{}$ respectively. Let the circle $\omega$ touch the segments $CA{}$ and $AB{}$ at $Q{}$ and $R{}$ respectively, and the points $M{}$ and $N{}$ are selected on the segments $AB{}$ and $AC{}$ respectively, so that the segments $CM{}$ and $BN{}$ touch $\omega$. The bisectors of $\angle NBC$ and $\angle MCB$ intersect the segments $DE{}$ and $DF{}$ at $K{}$ and $L{}$ respectively. Prove that the lines $RK{}$ and $QL{}$ intersect on $\omega$. [i]Proposed by Tran Quang Hung[/i]

Show that there are infinitely many rational triples $(a, b, c)$ such that $$a + b + c = abc = 6.$$

Let $P$ be a point on circumcircle of triangle $ABC$ on arc $\stackrel{\frown}{BC}$ which does not contain point $A$. Let lines $AB$ and $CP$ intersect at point $E$, and lines $AC$ and $BP$ intersect at $F$. If perpendicular bisector of side $AB$ intersects $AC$ in point $K$, and perpendicular bisector of side $AC$ intersects side $AB$ in point $J$, prove that: ${\left(\frac{CE}{BF}\right)}^2=\frac{AJ\cdot JE}{AK \cdot KF}$

Let $ABCD$ be a trapezoid such that $AD\parallel BC$ and $|AB|=|BC|$. Let $E$ and $F$ be the midpoints of $[BC]$ and $[AD]$, respectively. If the internal angle bisector of $\triangle ABC$ passes through $F$, find $|BD|/|EF|$.

Consider five spheres with radius $10$ cm . Four of these spheres are arranged on a horizontal table so that its centers form a $20$ cm square and the fifth sphere is placed on them so that it touches the other four. What is the distance between center of this fifth sphere and the table?

Let $a,m$ be positive integers such that $Ord_m (a)$ is odd and for any integers $x,y$ so that 1.$xy \equiv a \pmod m$ 2.$Ord_m(x) \le Ord_m(a)$ 3.$Ord_m(y) \le Ord_m(a)$ We have either $Ord_m(x)|Ord_m(a)$ or $Ord_m(y)|Ord_m(a)$.prove that $Ord_m(a)$ contains at most one prime factor.

$a,b,c\in\mathbb{R^+}$ and $a^2+b^2+c^2=48$. Prove that \[a^2\sqrt{2b^3+16}+b^2\sqrt{2c^3+16}+c^2\sqrt{2a^3+16}\le24^2\]

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]