Bài 64. For any integer d > 0, let f(d) be the smallest possible integer that has exactly d positive divisors (so for example we have f(1)=1, f(5)=16, and f(6)=12). Prove that for every integer k \geq 0 the number f\left(2^k\right) divides f\left(2^{k+1}\right).

Bài 65. Consider a polynomial $latex P(x)=\prod_{j=1}^9 (x+d_j),$ where d_1, d_2, \ldots d_9 are nine distinct integers. Prove that there exists an integer N, such that for all integers x \geq N the number P(x) is divisible by a prime number greater than 20.

Bài 66. For each positive integer k, let t(k) be the largest odd divisor of k. Determine all positive integers a for which there exists a positive integer n, such that all the differences

t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1) are divisible by 4.

Bài 67. Denote by \mathbb{Q}^+ the set of all positive rational numbers. Determine all functions f : \mathbb{Q}^+ \to \mathbb{Q}^+ which satisfy the following equation for all x, y \in \mathbb{Q}^+

f\left( f(x)^2y \right) = x^3 f(xy).

Bài 68. Find all functions g:\mathbb{N}^*\rightarrow\mathbb{N}^* such that

\left(g(m)+n\right)\left(g(n)+m\right) is perfect square for all m,n\in\mathbb{N}^*.

Bài 69. Determine all functions f from the set of positive integers to the set of positive integers such that, for all positive integers a and b, there exists a non-degenerate triangle with sides of lengths a, f(b) \text{ and } f(b + f(a) - 1).

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

Bài 70. Let n be a positive integer and let a_1,a_2,a_3,\ldots,a_k \left( k\ge 2\right) be distinct integers in the set \{ 1,2,\ldots,n\} such that n divides a_i(a_{i + 1} - 1) for i = 1,2,\ldots,k - 1. Prove that n does not divide a_k(a_1 - 1).

Bài 71. A positive integer N is called balanced, if N=1 or if N can be written as a product of an even number of not necessarily distinct primes. Given positive integers a and b, consider the polynomial P defined by P(x)=(x+a)(x+b).

(a) Prove that there exist distinct positive integers a and b such that all the number P(1), P(2),\ldots, P(50) are balanced.

(b) Prove that if P(n) is balanced for all positive integers n, then a=b.

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