Problem Section #1 a) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$, and y. Find the greatest possible value of: $x + y$. NOTE: There is a high chance that this problems was copied.

Problem Section #1 b) Let $a, b$ be positive integers such that $b^n +n$ is a multiple of $a^n + n$ for all positive integers $n$. Prove that $a = b.$

Problem Section #1 c) Find all pairs $(m, n)$ of non-negative integers for which $m^2+2.3^n=m(2^{n+1}-1).$

Problem Section #2 a) If $$ax+by=7$$$$ax^2+by^2=49$$$$ax^3+by^3=133$$$$ax^4+by^4=406$$, find the value of $2014(x+y-xy)-100(a+b).$

Problem Section #2 b) Find the maximal value of $(x^3+1)(y^3+1)$, where $x,y \in \mathbb{R}$, $x+y=1$.

Problem Section #2 c). Denote by $\mathbb{Q^+}$ the set of all positive rational numbers. Determine all functions $f:\mathbb{Q^+}\to\mathbb{Q^+}$ which satisfy the following equation for all $x,y \in \mathbb{Q^+} : f(f(x)^2.y)=x^3.f(xy)$.

Problem Section #3 a) Circles $O_1$ and $O_2$ interest at two points $B$ and $C$, and $BC$ is the diameter of circle $O_1$. Construct a tangent line of circle $O_1$ at $C$ and intersecting circle $O_2$ at another point $A$. Join $AB$ to intersect circle $O_1$ at point $E$, then join $CE$ and extend it to intersect circle $O_2$ at point $F$. Assume $H$ is an arbitrary point on line segment $AF$. Join $HE$ and extend it to intersect circle $O_1$ at point $G$, and then join $BG$ and extend it to intersect the extend of $AC$ at point $D$. Prove: $\frac{AH}{HF}=\frac{AC}{CD}$.

Problem Section #3 NOTE: Neglect that HF and CD.

Problem Section #3 c) Let $ABCDE$ be a convex pentagon such that $BC \parallel AE, AB = BC + AE$, and $\angle{ABC} =\angle{CDE}$. Let $M$ be the midpoint of $CE$, and let $O$ be the circumcenter of triangle $BCD$. Given that $\angle{DMO}=90^{o}$, prove that $2\angle{BDA} =\angle{CDE}$.

Problem Section #4 a) There is a $6 * 6$ grid, each square filled with a grasshopper. After the bell rings, each grasshopper jumps to an adjacent square (A square that shares a side). What is the maximum number of empty squares possible?

Problem Section #4 b) Let $A$ be a unit square. What is the largest area of a triangle whose vertices lie on the perimeter of $A$? Justify your answer.

Problem Section #4 c) A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. John will select a set of eight cards from the deck at random. Given that he gets at least one card of each color and at least one card with each number, the probability that John can discard one of his cards and still have at least one card of each color and at least one card with each number is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.