**Bài 64.** For any integer let be the smallest possible integer that has exactly positive divisors (so for example we have and ). Prove that for every integer the number divides

**Bài 65.** Consider a polynomial $latex P(x)=\prod_{j=1}^9 (x+d_j),$ where are nine distinct integers. Prove that there exists an integer such that for all integers the number is divisible by a prime number greater than

**Bài 66.** For each positive integer let be the largest odd divisor of Determine all positive integers for which there exists a positive integer such that all the differences

are divisible by .

**Bài 67.** Denote by the set of all positive rational numbers. Determine all functions which satisfy the following equation for all

**Bài 68.** Find all functions such that

is perfect square for all

**Bài 69.** Determine all functions from the set of positive integers to the set of positive integers such that, for all positive integers and , there exists a non-degenerate triangle with sides of lengths

(A triangle is non-degenerate if its vertices are not collinear.)

**Bài 70.** Let be a positive integer and let be distinct integers in the set such that divides for . Prove that does not divide

**Bài 71.** A positive integer is called *balanced*, if or if can be written as a product of an even number of not necessarily distinct primes. Given positive integers and , consider the polynomial defined by .

(a) Prove that there exist distinct positive integers and such that all the number , ,, are balanced.

(b) Prove that if is balanced for all positive integers , then .