SMO 2016 Senior Round 2

In a triangle $ABC$, $M$ is the midpoint of $BC$ and $D$ is the point on $BC$ such that $AD$ bisects $\angle BAC$. The line through $B$ perpendicular to $AD$ intersects $AD$ at $E$ and $AM$ at $G$. Prove that $GD$ is parallel to $AB$.

Let $n$ be a positive integer. Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly $n$ distinct points.

For any integer $n\ge1$, show that $$\sum_{k=1}^n\frac{2^k}{\sqrt{k+0.5}}\le2^{n+1}\sqrt{n+1}-\frac{4n^{3/2}}3.$$

Let $P$ be a 2016 sided polygon with all its adjacent sides perpendicular to each other, i.e., all its internal angles are either $90^\circ$ or $270^\circ$. If the lengths of its sides are odd integers, prove that its area is an even integer.

For each integer $n>1$, find a set of $n$ integers $\{a_1,a_2,\ldots,a_n\}$ such that the set of numbers $\{a_i+a_j\mid1\le i\le j\le n\}$ leave distinct remainders when divided by $n(n+1)/2$. If such a set of integers does not exist, give a proof.