Find the positive integer $ n$ such that \[\arctan\frac{1}{3}\plus{}\arctan\frac{1}{4}\plus{}\arctan\frac{1}{5}\plus{}\arctan\frac{1}{n}\equal{}\frac{\pi}{4}.\]

Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]

For a set of points in a plane, we construct the perpendicular bisectors of the line segments connecting every pair of those points and we count the number of points in which these perpendicular bisectors intersect each other. If we start with twelve points, the maximum possible number of intersection points is 1705. What is the maximum possible number of intersection points if we start with thirteen points?

Find the sum of the values of $x$ for which $\binom{x}{0}-\binom{x}{1}+\binom{x}{2}-...+\binom{x}{2008}=0$

Circle $\omega$ has radius 10 with center $O$. Let $P$ be a point such that $PO=6$. Let the midpoints of all chords of $\omega$ through $P$ bound a region of area $R$. Find the value of $\lfloor 10R \rfloor$. [i]Proposed by Andrew Zhao[/i]

The polynomial $f(x)=x^3-3x^2-4x+4$ has three real roots $r_1$, $r_2$, and $r_3$. Let $g(x)=x^3+ax^2+bx+c$ be the polynomial which has roots $s_1$, $s_2$, and $s_3$, where $s_1=r_1+r_2z+r_3z^2$, $s_2=r_1z+r_2z^2+r_3$, $s_3=r_1z^2+r_2+r_3z$, and $z=\frac{-1+i\sqrt3}2$. Find the real part of the sum of the coefficients of $g(x)$.

Let \[\begin{array}{ccccccccccc}A&=&5\cdot 6&-&6\cdot 7&+&7\cdot 8&-&\cdots&+&2003\cdot 2004,\\B&=&1\cdot 10&-&2\cdot 11&+&3\cdot 12&-&\cdots&+&1999\cdot 2008.\end{array}\] Find the value of $A-B$.

A toy factory produces several kinds of clay toys. The toys are painted in $k$ colours. [i]Diversity[/i] of a colour is the number of [i]different[/i] toys of that colour. (Thus, if there are $5$ blue cats, $7$ blue mice and nothing else is blue, the diversity of colour blue is $2$.) The painting protocol requires that [i]each colour is used and the diversities of each two colours are different[/i]. The toys in the store could be painted according to the protocol. However, a batch of clay Cheburashkas arrived at the store before painting (there were no Cheburashkas before). The number of Cheburashkas is not less that the number of the toys of any other kind. The total number of all toys, including Cheburashkas, is at least $\frac{(k+1)(k+2)}{2}$. Prove that now the toys can be painted in $k + 1$ colours according to the protocol. [i]Proposed by F. Petrov[/i]

The chords $AC$ and $BD$ of a, circle with centre $O$ intersect at the point $K$. The circumcentres of triangles $AKB$ and $CKD$ are $M$ and $N$ respectively. Prove that $OM = KN$. (A Zaslavsky )

Let f be a rational function (i.e. the quotient of two real polynomials) and suppose that $f(n)$ is an integer for infinitely many integers n. Prove that f is a polynomial.

Suppose that $x$, $y$, and $z$ are complex numbers such that $xy = -80-320i$, $yz = 60$, and $zx = -96+24i$, where $i = \sqrt{-1}$. Then there are real numbers $a$ and $b$ such that $x+y+z = a+bi$. Find $a^2 + b^2$.

A sheet of paper has the shape of an isosceles trapezoid $C_1AB_2C_2$ with the shorter base $B_2C_2$. The foot of the perpendicular from the midpoint $D$ of $C_1C_2$ to $AC_1$ is denoted by $B_1$. Suppose that upon folding the paper along $DB_1, AD$ and $AC_1$ points $C_1,C_2$ become a single point $C$ and points $B_1,B_2$ become a point $B$. The area of the tetrahedron $ABCD$ is $64$ cm$^3$ . Find the sides of the initial trapezoid.

Prove that for arbitary positive integer $ n\geq 4$, there exists a permutation of the subsets that contain at least two elements of the set $ G_{n} \equal{} \{1,2,3,\cdots,n\}$: $ P_{1},P_{2},\cdots,P_{2^n \minus{} n \minus{} 1}$ such that $ |P_{i}\cap P_{i \plus{} 1}| \equal{} 2,i \equal{} 1,2,\cdots,2^n \minus{} n \minus{} 2.$

Let $A$ be a set of positive integers with at least 2 elements. It is given that for any numbers $a>b$, $a,b \in A$ we have $\frac{ [a,b] }{ a- b } \in A$, where by $[a,b]$ we have denoted the least common multiple of $a$ and $b$. Prove that the set $A$ has [i]exactly[/i] two elements. [i]Marius Gherghu, Slatina[/i]

How many non-congruent scalene triangles with perimeter $21$ have integer side lengths that form an arithmetic sequence? (In an arithmetic sequence, successive terms differ by the same amount.) $\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }3\qquad\text{(D) }4\qquad\text{(E) }6$

In $\triangle ABC$, $AB > AC$. Point $P$ is on line $BC$ such that $AP$ is tangent to its circumcircle. Let $M$ be the midpoint of $AB$, and suppose the circumcircle of $\triangle PMA$ meets line $AC$ again at $N$. Point $Q$ is the reflection of $P$ with respect to the midpoint of segment $BC$. The line through $B$ parallel to $QN$ meets $PN$ at $D$, and the line through $P$ parallel to $DM$ meets the circumcircle of $\triangle PMB$ again at $E$. Show that the lines $PM$, $BE$, and $AC$ are concurrent.

Let $ABC$ be a triangle with incenter $I$ whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$. Point $P$ lies on $\overline{EF}$ such that $\overline{DP} \perp \overline{EF}$. Ray $BP$ meets $\overline{AC}$ at $Y$ and ray $CP$ meets $\overline{AB}$ at $Z$. Point $Q$ is selected on the circumcircle of $\triangle AYZ$ so that $\overline{AQ} \perp \overline{BC}$. Prove that $P$, $I$, $Q$ are collinear.

Let $n \ge 2$ be a positive integer and $\mathcal{F}$ the set of functions $f:\{1,2,\ldots,n\} \to \{1,2,\ldots,n\}$ that satisfy $f(k) \le f(k+1) \le f(k)+1,$ for all $k \in \{1,2,\ldots,n-1\}.$ a) Find the cardinal of the set $\mathcal{F}.$ b) Find the total number of fixed points of the functions in $\mathcal{F}.$

Find all natural numbers $n \ge 1$ such that there is a polynomial $p(x)$ with integer coefficients for which $p (1) = p (2) = 0$ and where $p (n)$ is a prime number .

Evaluate $ \int_0^{\frac{\pi}{2}} e^{xe^x}\{(x\plus{}1)e^x(\cos x\plus{}\sin x)\plus{}\cos x\minus{}\sin x\}dx$.

For a function $ f(x) \equal{} 6x(1 \minus{} x)$, suppose that positive constant $ c$ and a linear function $ g(x) \equal{} ax \plus{} b\ (a,\ b: \text{constants}\,\ a > 0)$ satisfy the following 3 conditions: $ c^2\int_0^1 f(x)\ dx \equal{} 1,\ \int_0^1 f(x)\{g(x)\}^2\ dx \equal{} 1,\ \int_0^1 f(x)g(x)\ dx \equal{} 0$. Answer the following questions. (1) Find the constants $ a,\ b,\ c$. (2) For natural number $ n$, let $ I_n \equal{} \int_0^1 x^ne^x\ dx$. Express $ I_{n \plus{} 1}$ in terms of $ I_n$. Then evaluate $ I_1,\ I_2,\ I_3$. (3) Evaluate the definite integrals $ \int_0^1 e^xf(x)\ dx$ and $ \int_0^1 e^xf(x)g(x)\ dx$. (4) For real numbers $ s,\ t$, define $ J \equal{} \int_0^1 \{e^x \minus{} cs \minus{} tg(x)\}^2\ dx$. Find the constants $ A,\ B,\ C,\ D,\ E$ by setting $ J \equal{} As^2 \plus{} Bst \plus{} Ct^2 \plus{} Ds\plus{}Et \plus{} F$. (You don't need to find the constant $ F$). (5) Find the values of $ s,\ t$ for which $ J$ is minimal.

A chess tournament had 10 participants. Each round, the participants split into pairs, and each pair played a game. In total, each participant played with every other participant exactly once, and in at least half of the games both the players were from the same town. Prove that during each round there was a game played by two participants from the same town. [i](Boris Frenkin)[/i]

Alex has a piece of cheese. He chooses a positive number a and cuts the piece into several pieces one by one. Every time he choses a piece and cuts it in the same ratio $1 : a$. His goal is to divide the cheese into two piles of equal masses. Can he do it if $(a) a$ is irrational? $(b) a$ is rational, $a \neq 1?$

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.

There are $28$ students who have to be separated into two groups such that the number of students in each group is a multiple of $4$. The number of ways to split them into the groups can be written as $$\sum_{k \ge 0} 2^k a_k = a_0 +2a_1 +4a_2 +...$$ where each $a_i$ is either $0$ or $1$. Find the value of $$\sum_{k \ge 0} ka_k = 0+ a_1 +2a_2 +3a3_ +....$$