SMO 2016 Junior Round 2

Find all integers $n$ such that $n^2+24n+35$ is a square.

Let $a_1,a_2,\ldots,a_9$ be a sequence of numbers satisfying $0\lt p\le a_i\le q$ for each $i=1,2,\ldots,9$. Prove that $$\frac{a_1}{a_9}+\frac{a_2}{a_8}+\cdots+\frac{a_9}{a_1}\le1+\frac{4(p^2+q^2)}{pq}.$$

In the triangle $ABC$, $\angle A=90^\circ$, the bisector of $\angle B$ meets the altitude $AD$ at the point $E$, and the bisector of $\angle CAD$ meets the side $CD$ at $F$. The line through $F$ perpendicular to $BC$ intersects $AC$ at $G$. Prove that $B,E,G$ are collinear.

A group of tourists get on 10 buses in the outgoing trip. The same group of tourists get on 8 buses in the return trip. Assuming each bus carries at least 1 tourist, prove that there are at least 3 tourists such that each of them has taken a bus in the return trip that has more people than the bus he has taken in the outgoing trip.

Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly 200 distinct points.

(Note that for 3 distinct points, the minimum number of lines is 3 and for 4 distinct points, the minimum is 4.)