Let $ \theta_1, \theta_2,\ldots , \theta_{2008}$ be real numbers. Find the maximum value of $ \sin\theta_1\cos\theta_2 + \sin\theta_2\cos\theta_3 + \ldots + \sin\theta_{2007}\cos\theta_{2008} + \sin\theta_{2008}\cos\theta_1$

Find the total number of solutions to the following system of equations: $ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37} \\ b(a + d)\equiv b \pmod{37} \\ c(a + d)\equiv c \pmod{37} \\ bc + d^2\equiv d \pmod{37} \\ ad - bc\equiv 1 \pmod{37} \end{array}$

Let $ ABCDE$ be an arbitrary convex pentagon. Suppose that $ BD\cap CE = A'$, $ CE\cap DA = B'$, $ DA\cap EB = C'$, $ EB\cap AC = D'$ and $ AC\cap BD = E'$. Suppose also that $ eABD'\cap eAC'E = A''$, $ eBCE'\cap eBD'A = B''$, $ eCDA'\cap eCE'B = C''$, $ eDEB'\cap eDA'C = D''$, $ eEAC'\cap eEB'D = E''$. Prove that $ AA'', BB'', CC'', DD'', EE''$ are concurrent. (Here $ l_1\cap l_2 = P$ means that $ P$ is the intersection of lines $ l_1$ and $ l_2$. Also $ eA_1A_2A_3\cap eB_1B_2B_3 = Q$ means that $ Q$ is the intersection of the circumcircles of $ \Delta A_1A_2A_3$ and $ \Delta B_1B_2B_3$.)

In a school there are 2008 students. Students are members of certain committees. A committee has at most 1004 members and every two students join a common committee. (a) Determine the smallest possible number of committees in the school. (b) If it is further required that the union of any two committees consists of at most 1800 students, will your answer in (a) still hold?

Let $ a,b,c$ be the three sides of a triangle. Determine all possible values of $ \frac {a^2 + b^2 + c^2}{ab + bc + ca}$

Show that the equation $ y^{37}\equiv x^3+11 \pmod p$ is solvable for every prime $ p$, where $ p\leq100$.

Let $ f: Z \to Z$ be such that $ f(1) = 1, f(2) = 20, f(-4) = -4$ and $ f(x+y) = f(x) +f(y)+axy(x+y)+bxy+c(x+y)+4 \forall x,y \in Z$, where $ a,b,c$ are constants. (a) Find a formula for $ f(x)$, where $ x$ is any integer. (b) If $ f(x) \geq mx^2+(5m+1)x+4m$ for all non-negative integers $ x$, find the greatest possible value of $ m$.

Prove that there are infinitely many primes $ p$ such that the total number of solutions mod $ p$ to the equation $ 3x^{3}+4y^{4}+5z^{3}-y^{4}z \equiv 0$ is $ p^2$

Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB = 90^{\circ}$. (a) Show that all such lines $ AB$ are concurrent. (b) Find the locus of midpoints of all such segments $ AB$.