N1.Find all pairs $(k,n)$ of positive integer for which $7^k-3^n$ divides $k^4+n^2$.
จะทยอยให้ครับ
N2.Let $b,n>1$ be integers. Suppose that for each $k>1$ there exists an integer $a_k $ such that $b-{a_k}^n$ is divisible by $k$. Prove that $b=A^n$ for some integer $A$.
N3.Let $X$ be a set of 10000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $Y$ of $X$ such that $a-b+c-d+e$ is not divisible by 47 for any $a,b,c,d,e \in Y$
N4.For every integer $k \ge 2$,prove that $2^{3k}$ divides the number
$\left( \matrix{
2^{k + 1} \cr
2^k \cr} \right) - \left( \matrix{
2^k \cr
2^{k - 1} \cr} \right)$
but $2^{3k+1}$ does not.
N5. Find all surjective functions $f:\mathbb{N}\rightarrow \mathbb{N} $ such that for every $m,n \in \mathbb{N} $ and every prime $p$,the number $f(m+n)$ is divisible by $p$ if and only if $f(m)+f(n)$ is divisible by $p$
N6. Let $k$ be a positive integer. Prove that the number $(4k^2-1)^2$ has a positive divisor of the form $8kn-1$ if and only if $k$ is even.
ปล. ข้อนี้เป็นข้อที่ใช้ในการแข่งขัน IMO 2007 แต่ดัดแปลงโจทย์เล็กน้อย
N7. For a prime $p$ and a positive integer $n$,denote by $v_p(n)$ the exponent of $p$ in the prime factorization of $n!$. Given a positive integer $d$ and a finite set $\{p_1,...p_k\}$ of primes. Show that there are infinitely many positive integers $n$ such that $d|v_{p_i}(n)$ for all $1 \le i \le k$