## Math Homework Problems

These are the two problems I assigned to the Senior students in the Malaysia IMO 2010 training camp. They are given two weeks to find the solutions.

1. Let $a_1, a_2, \dotsc, a_k$ be distinct positive integers not exceeding $n$ such that $a_i$ does not divide $\prod_{j \neq i} a_j$ for all $i$. Denote by $\pi(n)$ the number of primes not exceeding $n$. Prove that $k \leq \pi(n)$.

2. Let $f$ be a polynomial with rational coefficients such that $f(n) \in \mathbb{Z}$ for all $n \in \mathbb{Z}$. Prove that for any pair of distinct integers $m$ and $n$, the number

$\displaystyle \text{lcm}[1,2,\dotsc,\text{deg}(f)] \cdot \frac{f(m)-f(n)}{m-n}$

is an integer.

($\mathbb{Z}$ denotes the set of all integers, $\text{lcm}$ denotes the least common multiple, $\text{deg}(f)$ denotes the degree of polynomial $f$.)

Source: T. Andreescu & G. Dospinescu, Problems from the Book, pp. 69 & 228, with minor rewording.