Given the following list of numbers: $$1990, 1991, 1992, ..., 2002, 2003, 2003, 2003, ..., 2003$$where the number $2003$ appears $12$ times. Is it possible to write these numbers in some order so that the $100$-digit number that we get is prime?

Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$

A $4 \times 4$ board has all its squares painted white. An allowed operation is to choose a rectangle that contains $3$ squares and paint each of the boxes as follows: a) If the box is white then it is painted black. b) If the box is black then it is painted white. Prove that by applying the allowed operation several times, it is impossible get the entire board painted black.

The roots of the equation $x^2 + (3a + b)x + a^2 + 2b^2 = 0$ are $x_1$ and $x_2$ with $x_1 \ne x_2$. Determine the values of $a$ and $b$ so that the roots of the equation $ x^2 - 2a(3a + 2b)x + 5a^2b^2 + 4b^4 = 0$ let $x^2_1$ and $x^2_2$.

Prove that if $$\frac{p}{q}=1-\frac{1}{2} + \frac{1}{3}- \frac{1}{4} + ... -\frac{1}{1334} + \frac{1}{1335}$$where $p, q \in Z_+$ then $p$ is divisible by $2003$.

Let $ABC$ be an acute triangle and $T$ be a point interior to this triangle. that $\angle ATB = \angle BTC = \angle CTA$. Let $M,N$ and $P$ be the feet of the perpendiculars from $T$ to $BC$, $CA$ and $AB$ respectively. Prove that if the circle circumscribed around $\vartriangle MNP$ cuts again the sides $ BC$, $CA$ and $AB$ in $M_1$, $N_1$, $P_1$ respectively, then the $\vartriangle M_1N_1P_1$ It is equilateral.

Let $f : N \to N$ such that $f(p) = 1$ for all p prime and $f(ab) =bf(a) + af(b)$ for all $a, b \in N$. Prove that if $n = p^{a_1}_1 p^{a_1}_2... p^{a_1}_k$ is the canonical distribution of $n$ and $p_i$ does not divide $a_i$ ($i = 1, 2, ..., k$) then $\frac{n}{gcd(n,f(n))}$ is square free (not divisible by a square greater than $1$).

Let $a_1, a_2, ..., a_9$ be non-negative real numbers such that $a_1 = a_9 = 0$ and at least one of the remaining terms is different from $0$. a) Prove that for some $i$ $(i = 2, ..., 8$) ,holds that $a_{i-1} + a_{i+1} < 2a_i.$ b) Will the previous statement be true, if we change the number $2$ for $1.9$ in the inequality?

Let $P_1, P_2, P_3, P_4$ be four points on a circle, let $I_1$ be incenter of the triangle of vertices $P_2P_3P_4$, $I_2$ the incenter of the triangle $P_1P_3P_4$, $I_3$ the incenter of the triangle $P_1P_2P_4$, $I_4$ the incenter of the triangle $P_2P_3P_1$. Prove that $I_1I_2I_3I_4$ is a rectangle.

Let S(n) be the sum of the digits of the positive integer $n$. Determine $$S(S(S(2003^{2003}))).$$

Find all the functions $f : C \to R^+$ such that they fulfill simultaneously the following conditions: $$(i) \ \ f(uv) = f(u)f(v) \ \ \forall u, v \in C$$$$(ii) \ \ f(au) = |a | f(u) \ \ \forall a \in R, u \in C$$$$(iii) \ \ f(u) + f(v) \le |u| + |v| \ \ \forall u, v \in C$$

Let $D$ be the midpoint of the base $AB$ of the isosceles and acute angle triangle $ABC$, $E$ is a point on $AB$ and $O$ circumcenter of the triangle $ACE$. Prove that the line that passes through $D$ perpendicular to $DO$, the line that passes through $E$ perpendicular to $BC$ and the line that passes through$ B$ parallel to $AC$, they intersect at a point.