Find all the continuous bounded functions $f: \mathbb R \to \mathbb R$ such that \[(f(x))^2 -(f(y))^2 = f(x + y)f(x - y) \text{ for all } x, y \in \mathbb R.\]

Some cities of a country consisting of $n$ cities are connected by round trip flights so that there are at least $k$ flights from any city and any city is reachable from any city. Prove that for any such flight organization these flights can be distributed among $n-k$ air companies so that one can reach any city from any city by using of at most one flight of each air company.

An odd integer $ n \ge 3$ is said to be nice if and only if there is at least one permutation $ a_{1}, \cdots, a_{n}$ of $ 1, \cdots, n$ such that the $ n$ sums $ a_{1} \minus{} a_{2} \plus{} a_{3} \minus{} \cdots \minus{} a_{n \minus{} 1} \plus{} a_{n}$, $ a_{2} \minus{} a_{3} \plus{} a_{3} \minus{} \cdots \minus{} a_{n} \plus{} a_{1}$, $ a_{3} \minus{} a_{4} \plus{} a_{5} \minus{} \cdots \minus{} a_{1} \plus{} a_{2}$, $ \cdots$, $ a_{n} \minus{} a_{1} \plus{} a_{2} \minus{} \cdots \minus{} a_{n \minus{} 2} \plus{} a_{n \minus{} 1}$ are all positive. Determine the set of all `nice' integers.

Let $f$ be a function of the positive reals in the positive reals, such that $$f(x) \cdot f(y) - f(xy) = \frac{x}{y} + \frac{y}{x} \ \ for \ \ all \ \ x, y > 0 .$$ (a) Find $f(1)$. (b) Find $f(x)$.

$$\int_0^{20\pi}|x\sin(x)|\,dx$$ [i]Proposed by Connor Gordon[/i]

A hundred people want to take a photo. They can stand in any number of rows from 1 to 100. Let $N$ be the number of possible photos they can take. What is the largest integer $k$ such that $2^k \mid N$? [i]Lightning 2.4[/i]

Four positive integers $ a,b,c,$ and $ d$ have a product of 8! and satisfy\begin{align*}ab \plus{} a \plus{} b &\equal{} 524\\ bc \plus{} b \plus{} c &\equal{} 146\\ cd \plus{} c \plus{} d &\equal{} 104.\end{align*} What is $ a \minus{} d$? $ \textbf{(A)} \ 4 \qquad \textbf{(B)} \ 6 \qquad \textbf{(C)} \ 8 \qquad \textbf{(D)} \ 10 \qquad \textbf{(E)} \ 12$

Can one draw , on the surface of a Rubik's cube , a closed path which crosses each little square exactly once and does not pass through any vertex of a square? (S . Fomin, Leningrad)

Find all pairs $(x,y)$ of integers which satisfy the equation $(x + y)^2(x^2 + y^2) = 2009^2$

[b]a)[/b] Show that the set of nilpotents of a finite, commutative ring, is closed under each of the operations of the ring. [b]b)[/b] Prove that the number of nilpotents of a finite, commutative ring, divides the number of divisors of zero of the ring.

Find the value of $\frac{2014^3-2013^3-1}{2013\times 2014}$. $ \textbf{(A) }3\qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }11 $

In a convex quadrilateral $ABCD$ whose diagonals intersect at point $E$, the equalities$$\dfrac{|AB|}{|CD|}=\dfrac{|BC|}{|AD|}=\sqrt{\dfrac{|BE|}{|ED|}}$$hold. Prove that $ABCD$ is either a paralellogram or a cyclic quadrilateral

One day, before his work time at Jane Street, Sunny decided to have some fun. He saw that there are some real numbers $a_{-1},\ldots,a_{-k}$ on a blackboard, so he decided to do the following process just for fun: if there are real numbers $a_{-k},\ldots,a_{n-1}$ on the blackboard, then he computes the polynomial $$P_n(t)=(1-a_{-k}t)\cdots(1-a_{n-1}t).$$ He then writes a real number $a_n$, where $$a_n=\frac{iP_n(i)-iP_n(-i)}{P_n(i)+P_n(-i)}.$$ If $a_n$ is undefined (that is, $P_n(i)+P_n(-i)=0$), then he would stop and go to work. Show that if Sunny writes some real number on the blackboard twice (or equivalently, there exists $m>n\ge0$ such that $am=an$), then the process never stops. Moreover, show that in this case, all the numbers Sunny writes afterwards will already be written before. (usjl)

Let $\omega$ be the incircle of $\triangle ABC$ and $D,E,F$ be the tangency points on $BC ,CA, AB$. In $\triangle DEF$ let the altitudes from $E,F$ to $FD,DE$ intersect $AB, AC$ at $X ,Y$. Prove that the second intersection of $(AEX)$ and $(AFY)$ lies on $\omega$

[b]a)[/b] $ \lim_{n\to\infty } \sum_{j=1}^n\frac{n}{n^2+n+j} =1 $ [b]b)[/b] $ \lim_{n\to\infty } \left( n- \sum_{j=1}^n\frac{n^2}{n^2+n+j} \right) =3/2 $ [i]Cristinel Mortici[/i]

A circle $\omega$ centered at $O$ and a point $P$ inside it are given. Let $X$ be an arbitrary point of $\omega$, the line $XP$ and the circle $XOP$ meet $\omega$ for a second time at points $X_1$, $X_2$ respectively. Prove that all lines $X_1X_2$ are parallel.

A point $D$ is marked on the side $AC$ of triangle $ABC$. The circumscribed circle of triangle $ABD$ passes through the center of the inscribed circle of triangle $BCD$. Find $\angle ACB$ if $\angle ABC = 40^o$.

Prove that the remainder of dividing the square of an integer by $3$ is different from $2$.

Integers $ a,$ $ b,$ $ c,$ and $ d,$ not necessarily distinct, are chosen independantly and at random from $ 0$ to $ 2007,$ inclusive. What is the probability that $ ad \minus{} bc$ is even? $ \textbf{(A)}\ \frac {3}{8}\qquad \textbf{(B)}\ \frac {7}{16}\qquad \textbf{(C)}\ \frac {1}{2}\qquad \textbf{(D)}\ \frac {9}{16}\qquad \textbf{(E)}\ \frac {5}{8}$

Determine all positive integers $a$, $b$, $c$, $p$, where $p$ and $p+2$ are odd primes and \[2^ap^b=(p+2)^c-1.\]

Nine lines parallel to the base of a triangle divide the other sides each into $10$ equal segments and the area into $10$ distinct parts. If the area of the largest of these parts is $38$, then the area of the original triangle is $\textbf{(A) }180\qquad\textbf{(B) }190\qquad\textbf{(C) }200\qquad\textbf{(D) }210\qquad \textbf{(E) }240$

Let $F_0 = 0, F_{1} = 1$, and for $n \ge 1, F_{n+1} = F_n + F_{n-1}$. Define $a_n = \left(\frac{1 + \sqrt{5}}{2}\right)^n \cdot F_n$ . Then there are rational numbers $A$ and $B$ such that $\frac{a_{30} + a_{29}}{a_{26} + a_{25}} = A + B \sqrt{5}$. Find $A + B$.

Let $n$ be the answer to this problem. Hexagon $ABCDEF$ is inscribed in a circle of radius $90$. The area of $ABCDEF$ is $8n$, $AB = BC = DE = EF$, and $CD = FA$. Find the area of triangle $ABC$:

Let $d(n)$ be number of prime divisors of $n$. Prove that one can find $k,m$ positive integers for any positive integer $n$ such that $k-m=n$ and $d(k)-d(m)=1$

In a triangle $A_1A_2A_3$, the excribed circles corresponding to sides $A_2A_3$, $A_3A_1$, $A_1A_2$ touch these sides at $T_1$, $T_2$, $T_3$, respectively. If $H_1$, $H_2$, $H_3$ are the orthocenters of triangles $A_1T_2T_3$, $A_2T_3T_1$, $A_3T_1T_2$, respectively, prove that lines $H_1T_1$, $H_2T_2$, $H_3T_3$ are concurrent.