Let $O$ be the circumcenter of triangle $ABC$, and let $l$ be the line passing through the midpoint of segment $BC$ which is also perpendic...

1. Let $O$ be the circumcenter of triangle $ABC$, and let $l$ be the line passing through the midpoint of segment $BC$ which is also perpendicular to the bisector of angle $\angle BAC$. Suppose that the midpoint of segment $AO$ lies on $l$. Find $\angle BAC$.
2. Find all ordered triplets of positive integers $(a,\ b,\ c)$ such that $$2^a+3^b+1=6^c.$$
3. In a school, there are $n$ students and some of them are friends each other. (Friendship is mutual.) Define $a, b$ the minimum value which satisfies the following conditions
i) We can divide students into $a$ teams such that two students in the same team are always friends.
ii) We can divide students into $b$ teams such that two students in the same team are never friends.
Find the maximum value of $N = a+b$ in terms of $n$.
4. Let $\Gamma$ be the circumcircle of triangle $ABC$, and let $l$ be the tangent line of $\Gamma$ passing $A$. Let $D, E$ be the points each on side $AB, AC$ such that $BD : DA= AE : EC$. Line $DE$ meets $\Gamma$ at points $F, G$. The line parallel to $AC$ passing $D$ meets $l$ at $H$, the line parallel to $AB$ passing $E$ meets $l$ at $I$. Prove that there exists a circle passing four points $F, G, H, I$ and tangent to line $BC$.
5. Find the maximum value of real number $k$ such that $\frac{a}{1+9bc+k(b-c)^2}+\frac{b}{1+9ca+k(c-a)^2}+\frac{c}{1+9ab+k(a-b)^2}\geq \frac{1}{2}$ holds for all non-negative real numbers $a,\ b,\ c$ satisfying $a+b+c=1$.

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