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.

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