18/12/2012

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 $a=b$.