Let $a,b,c,d$ be distinct digits such that the product of the $2$-digit numbers $\overline{ab}$ and $\overline{cb}$ is of the form $\overline{ddd}$. Find all possible values of $a+b+c+d$.

Let $x$ be a real number such that $0<x<\frac{\pi}{2}$. Prove that \[\cos^2(x)\cot (x)+\sin^2(x)\tan (x)\ge 1\]

[u]Set 3[/u] [b]3.1[/b] Find the number of distinct values that can be made by inserting parentheses into the expression $$1\,\,\,\,\, - \,\,\,\,\, 1 \,\,\,\,\, -\,\,\,\,\, 1 \,\,\,\,\, - \,\,\,\,\, 1 \,\,\,\,\, - \,\,\,\,\, 1\,\,\,\,\, - \,\,\,\,\, 1$$ such that you don’t introduce any multiplication. For example, $(1-1)-((1-1)-1-1)$ is a valid way to insert parentheses, but $1 - 1(-1 - 1) - 1 - 1$ is not. [b]3.2[/b] Let $T$ be the answer from the previous problem. Katie rolls T fair 4-sided dice with faces labeled $0-3$. Considering all possible sums of these rolls, there are two sums that have the highest probability of occurring. Find the smaller of these two sums. [b]3.3[/b] Let $T$ be the answer from the previous problem. Amy has a fair coin that she will repeatedly flip until her total number of heads is strictly greater than her total number of tails. Find the probability she will flip the coin exactly T times. (Hint: Finding a general formula in terms of T is hard, try solving some small cases while you wait for $T$.) PS. You should use hide for answers.

Let $z$ be a positive integer that is not divisible by $8$. Furthermore, let $n \geqslant 2$ be a positive integer. Prove that none of the numbers of the form $z^n + z + 1$ is a square number. [i](Walther Janous)[/i]

A circle touches the sides $AB,BC, CD,DA$ of a square at points $K,L,M,N$ respectively, and $BU, KV$ are parallel lines such that $U$ is on $DM$ and $V$ on $DN$. Prove that $UV$ touches the circle.

(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]