SMO 2009 Senior Round 2

Let $M$ and $N$ be points on sides $AB$ and $AC$ of triangle $ABC$ respectively. If $$\frac{BM}{MA}+\frac{CN}{NA}=1,$$ show that $MN$ passes through the centroid of $ABC$.

Find all pairs of positive integers $n,m$ that satisfy the equation $3\cdot2^m+1=n^2$.

Let $A$ be an $n$-element subset of $\{1,2,\ldots,2009\}$ with the property that the difference between any two numbers in $A$ is not a prime number. Find the largest possible value of $n$. Find a set with this number of elements. (Note: $1$ is not a prime number.)

Let $a,b,c\gt 0$ such that $a+b+c=1$. Show that if $x_1,x_2,\ldots,x_5$ are positive real numbers such that $x_1x_2\cdots x_5=1$, then $$(ax_1^2+bx_1+c)(ax_2^2+bx_2+c)\cdots(ax_5^2+bx_5+c)\ge1.$$

In an archery competition, there are $30$ contestants. The target is divided into two zones. A hit in zone $1$ is awarded $10$ points while a hit in zone $2$ is awarded $5$ points. No point is awarded for a miss. Each contestant shoots $16$ arrows. At the end of the competition, statistics show that more than $50\%$ of the arrows hit zone $2$. The number of arrows that hit zone $1$ and the number that missed the target are equal. Prove that there are two contestants with the same score.