a) Calculate the product $$\left(1+\frac{1}{2}\right) \left(1+\frac{1}{3}\right) \left(1+\frac{1}{4}\right)... \left(1+\frac{1}{2006}\right) \left(1+\frac{1}{2007}\right)$$ b) Let the set $$A =\left\{\frac{1}{2}, \frac{1}{3},\frac{1}{4}, ...,\frac{1}{2006}, \frac{1}{2007}\right\}$$ Determine the sum of all products of $2$, of $4$, of $6$,... , of $2004$ ¸and of $ 2006$ different elements of the set $A$.

Consider $p$ a prime number and $p$ consecutive positive integers $m_{1}, m_{2}, \ldots, m_{p}$. Choose a permutation $\sigma$ of $1, 2, \ldots, p$. Show that there exist two different numbers $k,l \in \{1,2, \ldots, p\}$ such that $m_{k}m_{\sigma(k)}-m_{l}m_{\sigma(l)}$ is divisible by $p$.

Let $ ABC$ be a given triangle with the incenter $ I$, and denote by $ X$, $ Y$, $ Z$ the intersections of the lines $ AI$, $ BI$, $ CI$ with the sides $ BC$, $ CA$, and $ AB$, respectively. Consider $ \mathcal{K}_{a}$ the circle tangent simultanously to the sidelines $ AB$, $ AC$, and internally to the circumcircle $ \mathcal{C}(O)$ of $ ABC$, and let $ A^{\prime}$ be the tangency point of $ \mathcal{K}_{a}$ with $ \mathcal{C}$. Similarly, define $ B^{\prime}$, and $ C^{\prime}$. Prove that the circumcircles of triangles $ AXA^{\prime}$, $ BYB^{\prime}$, and $ CZC^{\prime}$ all pass through two distinct points.

A mother and her son are playing. At first, the son divides a ${300}\mathrm{\;g}$ wheel of cheese into 4 slices. Then the mother divides ${280}\mathrm{\;g}$ of butter between two plates. At last, the son puts the cheese slices on those plates. The son wins if on each plate the amount of cheese is not less than the amount of butter (otherwise the mother wins). Who of them can win irrespective of the opponent's actions? Alexandr Shapovalov

Two circles of radius 1 are to be constructed as follows. The center of circle $ A$ is chosen uniformly and at random from the line segment joining $ (0,0)$ and $ (2,0)$. The center of circle $ B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $ (0,1)$ to $ (2,1)$. What is the probability that circles $ A$ and $ B$ intersect? $ \textbf{(A)} \; \frac{2\plus{}\sqrt{2}}{4} \qquad \textbf{(B)} \; \frac{3\sqrt{3}\plus{}2}{8} \qquad \textbf{(C)} \; \frac{2 \sqrt{2} \minus{} 1}{2} \qquad \textbf{(D)} \; \frac{2\plus{}\sqrt{3}}{4} \qquad \textbf{(E)} \; \frac{4 \sqrt{3} \minus{} 3}{4}$

Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n))=n+1987$ for all $n$.

Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

Inside $\vartriangle ABC$, there is fixed a point $D$ such that $AC - DA > 1$ and $BC - BD > 1$. Prove that $EC - ED > 1$ for any point $E$ on segment $AB$.

$k,n$ are two arbitrary positive integers. Prove that there exists at least $(k-1)(n-k+1)$ positive integers that can be produced by $n$ number of $k$'s and using only $+,-,\times, \div$ operations and adding parentheses between them, but cannot be produced using $n-1$ number of $k$'s. [i]Proposed by Aryan Tajmir[/i]

Find the number of non-isosceles triangles (up to congruence) with integral side lengths, in which the sum of the two shorter sides is $19$.

4. There are 5 tables in a classroom. Each table has 4 chairs with a child sitting on it. All the children get up and randomly sit in a seat. Two people that sat at the same table before are not allowed to sit at the same table again. Assuming tables and chairs are distinguishable, if the number of different classroom arrangements can be written as $2^a3^b5^c$, what is $a+b+c$? [i]Proposed by Tragic[/i]

For a given integer $n \ge 2$, find the maximum possible value of $V_n = \sin x_1 \cos x_2 +\sin x_2 \cos x_3 +...+\sin x_n \cos x_1$, where $x_1,x_2,...,x_n$ are real numbers.

In the diagram below, rectangle $ABCD$ has $AB = 5$ and $AD = 12$. Also, $E$ is a point in the same plane outside $ABCD$ such that the perpendicular distances from $E$ to the lines $AB$ and $AD$ are $12$ and $1$, respectively, and $\triangle ABE$ is acute. There exists a line passing through $E$ which splits $ABCD$ into two figures of equal area. Suppose that this line intersects $\overline{AB}$ at a point $F$ and $\overline{CD}$ at a point $G$. Find $FG^2$. [asy] size(6.5cm); pair B=(0,0), C=(12,0), D=(12,5), A=(0,5); pair E=(-12,4); draw(A--E--B--C--D--cycle); draw(A--B); dot("$A$", A, NW); dot("$B$", B, SW); dot("$C$", C, SE); dot("$D$", D, NE); dot("$E$", E, W); [/asy] [i]Proposed by ApraTrip[/i]

Two circles, $C$ and $K$, are secant at $A$ and $B$. Let $P$ be a point on the arc $AB$ of $C$. Lines $PA$ and $PB$ intersect $K$ again at $R$ and $S$ respectively. Let $P'$ be another point at same arc as $P$, so that lines $P'A$ and $P'B$ again intersect $K$ at $R'$ and $S'$, respectively. Prove that the arcs $RS$ and $R'S'$ have equal measures. [img]https://cdn.artofproblemsolving.com/attachments/2/4/88693c36159179fb2b098b671a2f8281b37aae.png[/img]

For any positive integer $ k$, let $ f_k$ be the number of elements in the set $ \{ k \plus{} 1, k \plus{} 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s. (a) Prove that for any positive integer $ m$, there exists at least one positive integer $ k$ such that $ f(k) \equal{} m$. (b) Determine all positive integers $ m$ for which there exists [i]exactly one[/i] $ k$ with $ f(k) \equal{} m$.

$AD$ is the altitude on side $BC$ of triangle $ABC$. If $BC+AD-AB-AC = 0$, find the range of $\angle BAC$. [i]Alternative formulation.[/i] Let $AD$ be the altitude of triangle $ABC$ to the side $BC$. If $BC+AD=AB+AC$, then find the range of $\angle{A}$.

From the set of integers $ \{1,2,3,\ldots,2009\}$, choose $ k$ pairs $ \{a_i,b_i\}$ with $ a_i<b_i$ so that no two pairs have a common element. Suppose that all the sums $ a_i\plus{}b_i$ are distinct and less than or equal to $ 2009$. Find the maximum possible value of $ k$.

Determine a subset $ A\subset \mathbb{N}^*$ having $ 5$ different elements, so that the sum of the squares of its elements equals their product. Do not simply post the subset, show how you found it.

Numbers from $1$ to $100$ are written on the board. Is it possible to cross $10$ numbers in such way, that we couldn't select 10 numbers from rest which would form arithmetic progression?

Let $P(x)$ be a polynomial of degree $10$ with non-negative integer coefficients. The remainder when $P(x)$ is divided by $(x - 1)$ is $3$. How many such polynomials are there ?

Prove that if $$x^4_0+ a_1x^3_0+ a_2x^2_0+ a_3x_0 + a_4 = 0 \ \ and \ \ 4x^3_0+ 3a_1x^2_0+ 2a_2x_0 + a_3 = 0,$$ then $x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 $ is a mutliple of $(x - x_0)^2$.

Solve the system of equations: $$\begin{cases} \dfrac{xy}{x+y}=\dfrac{12}{5}\\ \\ \dfrac{yz}{y+z}=\dfrac{18}{5} \\ \\ \dfrac{zx}{z+y}=\dfrac{36}{13} \end{cases}$$

Given a circle $\Gamma$ with center at point $O$ and diameter $AB$. $OBDE$ is square, $F$ is the second intersection point of the line $AD$ and the circle $\Gamma$, $C$ is the midpoint of the segment $AF$. Find the value of the angle $OCB$.

There are $n$ students $ A_1 , A_2 , \cdots , A_n $ and some of them shaked hands with each other. ($ A_i $ and $ A_j$ can shake hands more than one time.) Let the student $ A_i $ shaked hands $ d_i $ times. Suppose $ d_1 + d_2 + \cdots + d_n > 0 $. Prove that there exist $ 1 \le i < j \le n $ satisfying the following conditions: (a) Two students $ A_i $ and $ A_j $ shaked hands each other. (b) $ \frac{(d_1 + d_2 + \cdots + d_n ) ^2 }{n^2 } \le d_i d_j $

A triangle $A_1B_1C_1$ is a parallel projection of a triangle $ABC$ in space. The parallel projections $A_1H_1$ and $C_1L_1$ of the altitude $AH$ and the bisector $CL$ of $\vartriangle ABC$ respectively are drawn. Using a ruler and compass, construct a parallel projection of : (a) the orthocenter, (b) the incenter of $\vartriangle ABC$.