SMO 2018
 

1. Let $O$ be the circumcenter of acute $\bigtriangleup ABC (AB<AC)$, the angle bisector of $B\hat A C$ meets $BC$ at $T$ and $M$ is the midpoint of $AT$. Point $P$ lies inside $\bigtriangleup ABC$ such that $PB\bot PC$. $D$, $E$ distinct from $P$ lies on the perpendicular to $AP$ through $P$ such that $BD=BP$, $CE=CP$. If $AO$ bisects segment $DE$, prove that $AO$ is tangent to the circumcircle of $\bigtriangleup AMP$.

2. Does there exist a set $A\subseteq \mathbb{N}^{\ast} $ such that for any positive integer $n$, $A\cap \left\{\,n,2n,3n,\ldots ,15n\right\} $ contains exactly one element? Please prove your conclusion.

3. Given a positive integer $m$. Let $A_{l}=(4l+1)(4l+2)\ldots \left(\,4(5^{m}+1)l\right) $ for any positive integer $l$. Prove that there exist infinite number of positive integer $l$ which $5^{5^{m}l}\mid A_{l}$ and $5^{5^{m}l+1}\nmid A_{l}$ and find the minimum value of $l$ satisfying the above condition.

4. There are $24$ participants attended a meeting. Each two of them shook hands once or not. A total of $216$ handshakes occured in the meeting. For any two participants who have shaken hands, at most $10$ among the rest $22$ participants have shaken hands with exactly one of these two persons. Define a friend circle to be a group of $3$ participants in which each person has shaken hands with the other two. Find the minimum possible value of friend circles.

2. เซตดังกล่าวมีจริงครับ และมีวิธี construct หลายวิธี ตัวที่ผมใช้ในห้องสอบคือ
$$A=\{ n\mid f(n)\equiv 0\pmod{15}\}$$
เมื่อ $f(n) = \nu_2(n) + 4\nu_3(n) + 9\nu_5(n) + 11\nu_7(n) + 7\nu_{11}(n) + 14\nu_{13}(n)$.

ฟังก์ชันนี้คืออะไรและมีที่มาอย่างไรครับ :please:

อยากทราบเกณฑ์การได้เหรียญแต่ละเหรียญครับ ต้องทำได้ประมาณกี่คะแนน พี่ๆ พอทราบไหมครับ