For real numbers $x$ and $y$ we define $M(x, y)$ to be the maximum of the three numbers $xy$, $(x- 1)(y - 1)$, and $x + y - 2xy$. Determine the smallest possible value of $M(x, y)$ where $x$ and $y$ range over all real numbers satisfying $0 \le x, y \le 1$.

Let $p(n)\geq 0$ for all positive integers $n$. Furthermore, $x(0)=0, v(0)=1$, and \[x(n)=x(n-1)+v(n-1), \qquad v(n)=v(n-1)-p(n)x(n) \qquad (n=1,2,\dots).\] Assume that $v(n)\to 0$ in a decreasing manner as $n \to \infty$. Prove that the sequence $x(n)$ is bounded if and only if $\sum_{n=1}^{\infty}n\cdot p(n)<\infty$.

For any $n,T \geq 2, n, T \in \mathbb{N}$, find all $a \in \mathbb{N}$ such that $\forall a_i > 0, i = 1, 2, \ldots, n$, we have \[\sum^n_{k=1} \frac{a \cdot k + \frac{a^2}{4}}{S_k} < T^2 \cdot \sum^n_{k=1} \frac{1}{a_k},\] where $S_k = \sum^k_{i=1} a_i.$

A $2008 \times 2009$ rectangle is divided into unit squares. In how many ways can you remove a pair of squares such that the remainder can be covered with $1 \times 2$ dominoes?

Find all triplets $(a, k, m)$ of positive integers that satisfy the equation $k + a^k = m + 2a^m$.

Let $m$ be a positive integer. Prove that there are integers $a, b, k$, such that both $a$ and $b$ are odd, $k\geq0$ and \[2m=a^{19}+b^{99}+k\cdot2^{1999}\]

For what values of $ n$ does there exist an $ n \times n$ array of entries -1, 0 or 1 such that the $ 2 \cdot n$ sums obtained by summing the elements of the rows and the columns are all different?

Let $p$ be a prime. It is given that there exists a unique nonconstant function $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$ such that $\chi(1) = 1$ and $\chi(mn) = \chi(m)\chi(n)$ for all $m, n \not\equiv 0 \pmod p$ (here the product $mn$ is taken mod $p$). For how many positive primes $p$ less than $100$ is it true that \[\sum_{a=1}^{p-1}a^{\chi(a)}\equiv 0\pmod p?\] Here as usual $a^{-1}$ denotes multiplicative inverse. [i]Proposed by David Altizio[/i]

$N$ numbers - ones and twos - are arranged in a circle. We mean a number formed by several digits arranged in a row (clockwise or counterclockwise). For what is the smallest value of $N$, all four-digit numbers whose writing contains only numbers $1$ and $2$, could they be among those shown?

Consider increasing integer sequences with elements from $1,\ldots,10^6$. Such a sequence is [i]Adriatic[/i] if its first element equals 1 and if every element is at least twice the preceding element. A sequence is [i]Tyrrhenian[/i] if its final element equals $10^6$ and if every element is strictly greater than the sum of all preceding elements. Decide whether the number of Adriatic sequences is smaller than, equal to, or greater than the number of Tyrrhenian sequences. (Proposed by Gerhard Woeginger, Austria)

Decide whether there exists a largest positive integer $n$ such that the inequality \[\frac{\frac{a^2}{b}+\frac{b^2}{a}}{2} \ge \sqrt[n]{\frac{a^n+b^n}{2}}\] holds for all positive real numbers $a$ and $b$. If such a largest positive integer $n$ exists, determine it.

$ABC$ is a triangle. $U$ is a point on the line $BC$. $I$ is the midpoint of $BC$. The line through $C$ parallel to $AI$ meets the line $AU$ at $E$. The line through $E$ parallel to $BC$ meets the line $AB$ at $F$. The line through $E$ parallel to $AB$ meets the line $BC$ at $H$. The line through $H$ parallel to $AU$ meets the line $AB$ at $K$. The lines $HK$ and $FG$ meet at $T. V$ is the point on the line $AU$ such that $A$ is the midpoint of $UV$. Show that $V, T$ and $I$ are collinear.

A contest began at noon one day and ended $ 1000$ minutes later. At what time did the contest end? \[ \textbf{(A)}\ 10: 00 \text{ p.m.} \qquad \textbf{(B)}\ \text{midnight} \qquad \textbf{(C)}\ 2: 30 \text{ a.m.} \\ \textbf{(D)}\ 4: 40 \text{ a.m.} \qquad \textbf{(E)}\ 6: 40 \text{ a.m.} \]

A jeweler covers the diagonal of a unit square with small golden squares in the following way: - the sides of all squares are parallel to the sides of the unit square - for each neighbour is their sidelength either half or double of that square (squares are neighbour if they share a vertex) - each midpoint of a square has distance to the vertex of the unit square equal to $\dfrac12, \dfrac14, \dfrac18, ...$ of the diagonal. (so real length: $\times \sqrt2$) - all midpoints are on the diagonal (a) What is the side length of the middle square? (b) What is the total gold-plated area? [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=281[/img]

Let $ABC$ be an acute triange with $B,C$ fixed. Let $D$ be the midpoint of $BC$ and $E,F$ be the foot of the perpendiculars from $D$ to $AB,AC$, respectively. a) Let $O$ be the circumcenter of triangle $ABC$ and $M=EF\cap AO, N=EF\cap BC$. Prove that the circumcircle of triangle $AMN$ passes through a fixed point; b) Assume that tangents of the circumcircle of triangle $AEF$ at $E,F$ are intersecting at $T$. Prove that $T$ is on a fixed line.

Let $O$ be the circumcenter of triangle $\triangle ABC$, where $\angle BAC=120^{\circ}$. The tangent at $A$ to $(ABC)$ meets the tangents at $B,C$ at $(ABC)$ at points $P,Q$ respectively. Let $H,I$ be the orthocenter and incenter of $\triangle OPQ$ respectively. Define $M,N$ as the midpoints of arc $\overarc{BAC}$ and $OI$ respectively, and let $MN$ meet $(ABC)$ again at $D$. Prove that $AD$ is perpendicular to $HI$.

Let $x=\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ}.$ What is the greatest integer that does not exceed $100x$?

Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b$, $1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves from $(a, b)$ to $(c, d)$ such that $(|a-c|, |b - d|)$ is either $(1, k)$ or $(k, 1)$. The $k$-knight starts at cell $(1, 1)$, and performs several moves. A sequence of moves is a sequence of cells $(x_0, y_0)= (1, 1)$, $(x_1, y_1), (x_2, y_2)$, $\ldots, (x_n, y_n)$ such that, for all $i = 1, 2, \ldots, n$, $1 \leq x_i , y_i \leq 100$ and the $k$-knight can move from $(x_{i-1}, y_{i-1})$ to $(x_i, y_i)$. In this case, each cell $(x_i, y_i)$ is said to be reachable. For each $k$, find $L(k)$, the number of reachable cells.

What is the value of the improper integral $ \int_0 ^ {\pi} \log (\sin (x)) dx$?

Find all 4-tuples $(a,b,c,n)$ of naturals such that $n^a + n^b = n^c$

Do there exist a rectangle that can be partitioned into a regular hexagon with side length $1$, and several right triangles with side lengths $1, \sqrt3 , 2$?

Anna and Lisa go for a bike ride. Anna's bike breaks down $30$ kilometers before their final destination. The two decide to complete the ride with Lisa's bike as follows: At the beginning, Anna is riding a bike and Lisa leaves. At some point, Anna gets off the bike, parks it on the side of the road and continues by foot. When Lisa gets to the bike, she takes it and rides until she catches up with Anna. After that, they repeat the same procedure. We don't know how many times the procedure is repeated, but they arrive at the final goal at the same time. Anna walks at a speed of $4$ km/h and cycles at a speed of $15$ km/h. Lisa walks at $5$ km/h and cycles with $20$ km/h. How long does it take them to cover the last $30$ km of the road? (Neglect the time it takes to park, lock, unlock the bike, etc.)

Find the number of positive integers less than 101 that [i]can not [/i] be written as the difference of two squares of integers.

Let $\{\omega_1,\omega_2,\cdots,\omega_{100}\}$ be the roots of $\frac{x^{101}-1}{x-1}$ (in some order). Consider the set $$S=\{\omega_1^1,\omega_2^2,\omega_3^3,\cdots,\omega_{100}^{100}\}.$$ Let $M$ be the maximum possible number of unique values in $S,$ and let $N$ be the minimum possible number of unique values in $S.$ Find $M-N.$

Find the sum of $1\cdot 1!+2\cdot 2!+3\cdot 3!+\cdots+(n-1)(n-1)!+n\cdot n!$, where $n!=n(n-1)(n-2)\cdots2\cdot1$.