khulna math club probs

[Tahmid Hasan]

1.9 points are inscribed in a square with side length 1.prove that that is at least 3 points which make a triangle with area less than $$\frac{1}{8}$$
2what is the largest factor of $$11!$$ which has remainder 1 if divided by 6?
3.if $$mn$$ divides $$m^2+n^2+m$$,prove that $$m$$ is a square number.(m and n are integers)
4.$$a_1,a_2,... $$ is a sequence of non-negative real numbers
for all n $$a_n+a_{2n} \geq 3n$$ and $$a_{n+1}+n \leq 2 \sqrt{a_n(n+1)}$$
prove that for all n $$a_n \geq n$$
[took my sweat out to post the last prob so at least read it 1ce ]

[Moon]

Nice problems:
2.

Let $d$ be the largest factor such that $d\equiv 1\pmod{6}$. First notice that $2,3$ is relatively prime to $d$. (Because we have $2\cdot 3 | (d-1)$. So $d$ is a factor of $5\cdot 7 \cdot 11$. And $5\cdot 7 \cdot 11 \equiv -1 \cdot 1 \cdot -1\pmod{6}. $