(i) Find the angles of $\triangle ABC$ if the length of the altitude through $B$ is equal to the length of the median through $C$ and the length of the altitude through $C$ is equal to the length of the median through $B$. (ii) Find all possible values of $\angle ABC$ of $\triangle ABC$ if the length of the altitude through $A$ is equal to the length of the median through $C$ and the length of the altitude through $C$ is equal to the length of the median through $B$.

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

Prove that for any $0<\delta <2\pi$ there exists a number $m>1$ such that for any positive integer $n$ and unimodular complex numbers $z_1,\ldots, z_n$ with $z_1^v+\dots+z_n^v=0$ for all integer exponents $1\le v\le m$, any arc of length $\delta$ of the unit circle contains at least one of the numbers $z_1,\ldots, z_n$.

Find all real positive solutions (if any) to \begin{align*} x^3+y^3+z^3 &= x+y+z, \mbox{ and} \\ x^2+y^2+z^2 &= xyz. \end{align*}

Consider the determinant of the matrix $(a_{ij})_{ij}$ with $1\leq i,j \leq 100$ and $a_{ij}=ij.$ Prove that if the absolute value of each of the $100!$ terms in the expansion of this determinant is divided by $101,$ then the remainder is always $1.$

Let $ABCDE$ be pentagon such that $AE = ED$ and $BC = CD$. It is known that $\angle BAE + \angle EDC + \angle CB A = 360^o$ and that $P$ is the midpoint of $AB$. Show that the triangle $ECP$ is right.

Assuming that $f(x)$ is continuous in the interval $(0,1)$, prove that $$\int_{x=0}^{x=1} \int_{y=x}^{y=1} \int_{z=x}^{z=y} f(x)f(y)f(z)\;dz dy dx= \frac{1}{6}\left(\int_{0}^{1} f(t)\; dt\right)^{3}.$$

Let $ M $ denote the set of the primitives of a function $ f:\mathbb{R}\longrightarrow\mathbb{R} . $ [b]ii)[/b] Show that $ M $ along with the operation $ *:M^2\longrightarrow M $ defined as $ F*G=F+G(2007) $ form a commutative group. [b]iii)[/b] Show that $ M $ is isomorphic with the additive group of real numbers.

Consider equation I: $x+y+z=46$ where $x, y,$ and $z$ are positive integers, and equation II: $x+y+z+w=46$, where $x, y, z,$ and $w$ are positive integers. Then $ \textbf{(A)}\ \text{I can be solved in consecutive integers} \qquad$ $\textbf{(B)}\ \text{I can be solved in consecutive even integers} \qquad$ $\textbf{(C)}\ \text{II can be solved in consecutive integers} \qquad$ $\textbf{(D)}\ \text{II can be solved in consecutive even integers} \qquad$ $\textbf{(E)}\ \text{II can be solved in consecutive odd integers} $

Determine the highest power of $2$ that divides $2^n!$.

Japanese businessman Rui lives in America and makes a living from trading cows. On Black Thursday he was selling his cows for $2000$ dollars each (the cows were of the same price), but after the financial crash there were huge fluctuations in the market and Rui was forced to follow them with his pricing. Every day he doubled, halved, multiplied by five or divided by five the price from the previous day (even if it meant he had to give change in cents). At the same time he managed to follow the Japanese superstition, so that the integer part of the price in dollars never started with digit $4$. On the day when Billy visited him to buy some cows the price of each cow was $80$ dollars. What is the minimal number of days that could have passed since Black Thursday by then?

Let $ABC$ and $A_{1}B_{1}C_{1}$ be two triangles. Prove that $\frac{a}{a_{1}}+\frac{b}{b_{1}}+\frac{c}{c_{1}}\leq\frac{3R}{2r_{1}}$, where $a = BC$, $b = CA$, $c = AB$ are the sidelengths of triangle $ABC$, where $a_{1}=B_{1}C_{1}$, $b_{1}=C_{1}A_{1}$, $c_{1}=A_{1}B_{1}$ are the sidelengths of triangle $A_{1}B_{1}C_{1}$, where $R$ is the circumradius of triangle $ABC$ and $r_{1}$ is the inradius of triangle $A_{1}B_{1}C_{1}$.

Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.

Let $n$ be the answer to this problem. Box $B$ initially contains n balls, and Box $A$ contains half as many balls as Box $B$. After $80$ balls are moved from Box $A$ to Box $B$, the ratio of balls in Box $A$ to Box $B$ is now $\frac{p}{q}$ , where $p$, $q$ are positive integers with gcd$(p, q) = 1$. Find $100p + q$.

If all $x_k$ ($k = 1, 2, 3, 4, 5)$ are positive reals, and $\{a_1,a_2, a_3, a_4, a_5\} = \{1, 2,3 , 4, 5\}$, find the maximum of $$\frac{(\sqrt{s_1x_1} +\sqrt{s_2x_2}+\sqrt{s_3x_3}+\sqrt{s_4x_4}+\sqrt{s_5x_5})^2}{a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 + a_5x_5}$$ ($s_k = a_1 + a_2 +... + a_k$)

A circular game board is divided into $n \ge 3$ sectors. Each sector is either empty or occupied by a marker. In each step one chooses an occupied sector, removes its marker and then switches each of the two adjacent sectors from occupied to empty or vice-versa. Starting with a single occupied sector, for which $n$ is it possible to end up with all empty sectors after finitely many steps?

The area of the figure enclosed by the $x$-axis, $y$-axis, and line $7x + 8y = 15$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that \[ {AM\over AC}={CN\over CE}=r. \] Determine $r$ if $B,M$ and $N$ are collinear.

For a distributive lattice $ L$, consider the following two statements: (A) Every ideal of $ L$ is the kernel of at least two different homomorphisms. (B) $ L$ contains no maximal ideal. Which one of these statements implies the other? (Every homomorphism $ \varphi$ of $ L$ induces an equivalence relation on $ L$: $ a \sim b$ if and only if $ a \varphi\equal{}b \varphi$. We do not consider two homomorphisms different if they imply the same equivalence relation.) [i]J. Varlet, E. Fried[/i]

Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers defined by $a_{1}=1$, $a_{2}=2$ and $$\frac{a_{n+1}^{4}}{a_{n}^3} = 2a_{n+2}-a_{n+1}.$$ Prove that the following inequality holds for every positive integer $N>1$: $$\sum_{k=1}^{N}\frac{a_{k}^{2}}{a_{k+1}}<3.$$ [i]Note: The bound is not sharp.[/i] [i]Authored by Nikola Velov[/i]

Let $ l_i,$ $ i \equal{} 1,2,3$ be three non-collinear straight lines in the plane, which build a triangle, and $ f_i$ the axial reflections in $ l_i$. Prove that for each point $ P$ in the plane there exists finite interconnections (compositions) of the reflections of $ f_i$ which carries $ P$ into the triangle built by the straight lines $ l_i,$ i.e. maps that point to a point interior to the triangle.

In a triangle with sides $a, b, c$ the side $a$ is the arithmetic mean of $b$ and $c$. Prove that: a) $0^o \le A \le 60^o$. b) The height relative to side $a$ is three times the inradius $r$. c) The distance from the circumcenter to side $a$ is $R - r$, where $R$ is the circumradius.

Given is a grid with $2$ rows and $120$ columns, such that each cell has a number from the set $1, 2, ..., 120$. It is known that in each column, the upper number in it is smaller than the lower number, and in each row, the numbers are in non-strict increasing order from left to right. Prove that the number of these tables is multiple of $239$.

If $\alpha$ and $\beta$ are the roots of the equation $3x^2+x-1=0$, where $\alpha>\beta$, find the value of $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}$. $ \textbf{(A) }\frac{7}{9}\qquad\textbf{(B) }-\frac{7}{9}\qquad\textbf{(C) }\frac{7}{3}\qquad\textbf{(D) }-\frac{7}{3}\qquad\textbf{(E) }-\frac{1}{9} $

Given points $A, B, C, D, E$, and $F$ on a line (not necessarily in that order) with $AB = 2$, $BC = 6$, $CD = 8$, $DE = 10$, $EF = 20$, and $FA = 22$. Find the distance between the two furthest points on the line.