Let $S$ be the set of all partitions of $2000$ (in a sum of positive integers). For every such partition $p$, we define $f (p)$ to be the sum of the number of summands in $p$ and the maximal summand in $p$. Compute the minimum of $f (p)$ when $p \in S .$

Prove or disprove: For any positive integer $k$ there exists an integer $n > 1$ such that the binomial coeffcient $\binom{n}{i}$ is divisible by $k$ for any $1 \leq i \leq n-1.$

Let ${ABC}$ be a non-equilateral triangle. The incircle is tangent to the sides ${BC,CA,AB}$ at ${A_1,B_1,C_1}$, respectively, and M is the orthocenter of triangle ${A_1B_1C_1}$. Prove that ${M}$ lies on the line through the incenter and circumcenter of ${\vartriangle ABC}$.

Let $A$ and $B$ be two subsets of $S = \{1, 2, . . . , 2000\}$ with $|A| \cdot |B| \geq 3999$. For a set $X$ , let $X-X$ denotes the set $\{s-t | s, t \in X, s \not = t\}$. Prove that $(A-A) \cap (B-B)$ is nonempty.

For a given integer $d$, let us define $S = \{m^{2}+dn^{2}| m, n \in\mathbb{Z}\}$. Suppose that $p, q$ are two elements of $S$ , where $p$ is prime and $p | q$. Prove that $r = q/p$ also belongs to $S$ .

Let $k$ and $l$ be two given positive integers and $a_{ij}(1 \leq i \leq k, 1 \leq j \leq l)$ be $kl$ positive integers. Show that if $q \geq p > 0$, then \[(\sum_{j=1}^{l}(\sum_{i=1}^{k}a_{ij}^{p})^{q/p})^{1/q}\leq (\sum_{i=1}^{k}(\sum_{j=1}^{l}a_{ij}^{q})^{p/q})^{1/p}.\]