Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that \[ \left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}. \] Prove that $c_{n}\equiv 1\pmod{3} $ if and only if $n\equiv 2\pmod{3}.$

For three points $A,B,C$ in the plane, we define $m(ABC)$ to be the smallest length of the three heights of the triangle $ABC$, where in the case $A$, $B$, $C$ are collinear, we set $m(ABC) = 0$. Let $A$, $B$, $C$ be given points in the plane. Prove that for any point $X$ in the plane, \[ m(ABC) \leq m(ABX) + m(AXC) + m(XBC). \]

Let $n,k$ be positive integers such that n is not divisible by 3 and $k \geq n$. Prove that there exists a positive integer $m$ which is divisible by $n$ and the sum of its digits in decimal representation is $k$.

A square matrix is sum-magic if the sum of all elements in each row, column and major diagonal is constant. Similarly, a square matrix is product-magic if the product of all elements in each row, column and major diagonal is constant. Determine if there exist $3\times 3$ matrices of real numbers which are both sum-magic and product-magic.

In acute triangle $ABC$, $D,E$ are two points on side $BC$, satisfying that $\angle BAE=\angle CAF$. $FM\perp AB,EN\perp AC$ ($M,N$ are foot points). $AE$ intersects the circumcircle of $\triangle ABC$ at $D$. Prove that the area of $\triangle ABC$ and quadrilateral $AMDN$ are equal.

An (unordered) partition $P$ of a positive integer $n$ is an $n$-tuple of nonnegative integers $P=(x_1,x_2,\cdots,x_n)$ such that $\sum_{k=1}^n kx_k=n$. For positive integer $m\leq n$, and a partition $Q=(y_1,y_2,\cdots,y_m)$ of $m$, $Q$ is called compatible to $P$ if $y_i\leq x_i$ for $i=1,2,\cdots,m$. Let $S(n)$ be the number of partitions $P$ of $n$ such that for each odd $m<n$, $m$ has exactly one partition compatible to $P$ and for each even $m<n$, $m$ has exactly two partitions compatible to $P$. Find $S(2010)$.

Kevin is running $1000$ meters. He wants to have an average speed of $10$ meters a second. He runs the first $100$ meters at a speed of $4$ meters a second. Compute how quickly, in meters per second, he must run the last $900$ meters to attain his desired average speed of $10$ meters a second.

Given an integer $n\geq 2,$ colour red exactly $n$ cells of an infinite sheet of grid paper. A rectangular grid array is called special if it contains at least two red opposite corner cells; single red cells and 1-row or 1-column grid arrays whose end-cells are both red are special. Given a configuration of exactly $n$ red cells, let $N$ be the largest number of red cells a special rectangular grid array may contain. Determine the least value $N$ may take over all possible configurations of exactly $n$ red cells

The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names? $\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad \textbf{(E) }7$ [asy] unitsize(0.9cm); draw((-0.5,0)--(10,0), linewidth(1.5)); draw((-0.5,1)--(10,1)); draw((-0.5,2)--(10,2)); draw((-0.5,3)--(10,3)); draw((-0.5,4)--(10,4)); draw((-0.5,5)--(10,5)); draw((-0.5,6)--(10,6)); draw((-0.5,7)--(10,7)); label("frequency",(-0.5,8)); label("0", (-1, 0)); label("1", (-1, 1)); label("2", (-1, 2)); label("3", (-1, 3)); label("4", (-1, 4)); label("5", (-1, 5)); label("6", (-1, 6)); label("7", (-1, 7)); filldraw((0,0)--(0,7)--(1,7)--(1,0)--cycle, black); filldraw((2,0)--(2,3)--(3,3)--(3,0)--cycle, black); filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, black); filldraw((6,0)--(6,4)--(7,4)--(7,0)--cycle, black); filldraw((8,0)--(8,4)--(9,4)--(9,0)--cycle, black); label("3", (0.5, -0.5)); label("4", (2.5, -0.5)); label("5", (4.5, -0.5)); label("6", (6.5, -0.5)); label("7", (8.5, -0.5)); label("name length", (4.5,-1.5)); [/asy]

''Moskvich'' and ''Zaporozhets'' drove past the observer on the highway and the Niva moving towards them. It is known that when the Moskvich caught up with the observer, it was equidistant from the Zaporozhets and the Niva, and when the Niva caught up with the observer, it was equal. but removed from ''Moskvich'' and ''Zaporozhets''. Prove that ''Zaporozhets'' at the moment of passing by the observer was equidistant from the Niva and ''Moskvich''.

Let $ABC$ be a triangle, $X$ the midpoint of arc $BC$ on the circumcircle. The tangents from $X$ to the incircle meet the circumcircle again at $X_1,X_2$, and $X_1X_2$ intersects the incircle at $P,Q$. Let $M$ be the midpoint of $PQ$, and let $A_1$ be the tangency point of the $A$-mixtillinear incircle with the circumcircle. Show that $A,M,A_1$ are collinear.

In a graph with $2n-1$ vertices throwing out any vertex the remaining graph has a complete subgraph with $n$ vertices. Prove that the initial graph has a complete subgraph with $n+1$ vertices.

Assume $f:\mathbb N_0\to\mathbb N_0$ is a function such that $f(1)>0$ and, for any nonnegative integers $m$ and $n$, $$f\left(m^2+n^2\right)=f(m)^2+f(n)^2.$$(a) Calculate $f(k)$ for $0\le k\le12$. (b) Calculate $f(n)$ for any natural number $n$.

Define $\left\lVert A-B \right\rVert = (x_A-x_B)^2+(y_A-y_B)^2$ for every two points $A = (x_A, y_A)$ and $B = (x_B, y_B)$ in the plane. Let $S$ be the set of points $(x,y)$ in the plane for which $x,y \in \left\{ 0,1,\dots,100 \right\}$. Find the number of functions $f : S \to S$ such that $\left\lVert A-B \right\rVert \equiv \left\lVert f(A)-f(B) \right\rVert \pmod{101}$ for any $A, B \in S$. [i] Proposed by Victor Wang [/i]

Inside a circle, point $K$ is taken such that the ray drawn from $K$ through the centre $O$ of the circle and the chord perpendicular to this ray passing through $K$ divide the circle into three pieces with equal area. Let $L$ be one of the endpoints of the chord mentioned. Does the inequality $\angle KOL < 75^o$ hold?

Let $S$ be the center of a square $ABCD$ and $P$ be the midpoint of $AB$. The lines $AC$ and $PD$ meet at $M$, and the lines $BD$ and $PC$ meet at $N$. Prove that the radius of the incircle of the quadrilateral $PMSN$ equals $MP-MS$.

How many ordered pairs $(a, b)$ of positive integers with $1 \le a, b \le 10$ are there such that in the geometric sequence whose first term is $a$ and whose second term is $b$, the third term is an integer?

Find all triples $(a, b, c)$ of positive integers with $a+b+c = 10$ such that there are $a$ red, $b$ blue and $c$ green points (all different) in the plane satisfying the following properties: $\bullet$ for each red point and each blue point we consider the distance between these two points, the sum of these distances is $37$, $\bullet$ for each green point and each red point we consider the distance between these two points, the sum of these distances is $30$, $\bullet$ for each blue point and each green point we consider the distance between these two points, the sum of these distances is $1$.

For \(u,v \in\mathbb{R}^4\), let \(<u,v>\) denote the usual dot product. Define a [i]vector field[/i] to be a map \(\omega:\mathbb{R}\to\mathbb{R}\) such that \(<\omega(z),z>=0,\ \forall z\in\mathbb{R}^4.\) Find a maximal collection of vector fields \(\left\{\omega_1,...,\omega_k\right\}\) such that the map \(\Omega\) sending \(z\) to \(\lambda_1\omega_1(z)+\cdots+\lambda_k \omega_k(z)\), with \(\lambda_1,\ldots,\lambda_k\in\mathbb{R}\), is nonzero on \(\mathbb{R}^4\backslash\{0\}\) unless \(\lambda_1=\cdots=\lambda_k=0\)

Let \(a_0, a_1, a_2, \dots\) be an infinite sequence of positive integers with the following properties: - \(a_0\) is a given positive integer; - For each integer \(n \geq 1\), \(a_n\) is the smallest integer greater than \(a_{n-1}\) such that \(a_n + a_{n-1}\) is a perfect square. For example, if \(a_0 = 3\), then \(a_1 = 6\), \(a_2 = 10\), \(a_3 = 15\), and so on. (a) Let \(T\) be the set of numbers of the form \(a_k - a_l\), with \(k \geq l \geq 0\) integers. Prove that, regardless of the value of \(a_0\), the number of positive integers not in \(T\) is finite. (b) Calculate, as a function of \(a_0\), the number of positive integers that are not in \(T\).

Determine whether there exists an infinite sequence $a_1, a_2, a_3, \dots$ of positive integers which satisfies the equality \[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \] for every positive integer $n$.

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

The number $5.6$ may be expressed uniquely (ignoring order) as a product $\underline{a}.\underline{b} \times \underline{c}.\underline{d}$ for digits $a,b,c,d$ all nonzero. Compute $\underline{a}.\underline{b}+\underline{c}.\underline{d}.$

Triangle $ABC$ has $AB=13$, $BC=14$, and $AC=15$. The points $D, E,$ and $F$ are the midpoints of $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$ respectively. Let $ X \ne E$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$. What is $XA+XB+XC$? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \qquad \textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad \textbf{(E)}\ \frac{69\sqrt{2}}{4} $

Given a convex polygon $ A_1A_2 \ldots A_n$ with area $ S$ and a point $ M$ in the same plane, determine the area of polygon $ M_1M_2 \ldots M_n,$ where $ M_i$ is the image of $ M$ under rotation $ R^{\alpha}_{A_i}$ around $ A_i$ by $ \alpha_i, i \equal{} 1, 2, \ldots, n.$