Georgina calls a $992$-element subset $A$ of the set $S = \{1, 2, 3, \ldots , 1984\}$ a [b]halfthink set[/b] if [list] [*] the sum of the elements in $A$ is equal to half of the sum of the elements in $S$, and [*] exactly one pair of elements in $A$ differs by $1$. [/list] She notices that for some values of $n$, with $n$ a positive integer between $1$ and $1983$, inclusive, there are no halfthink sets containing both $n$ and $n+1$. Find the last three digits of the product of all possible values of $n$. [i]Proposed by Andrew Wu and Jason Wang[/i] (Note: wording changed from original to specify what $n$ can be.)

Janet and Donald agree to meet for lunch between 11:30 and 12:30. They each arrive at a random time in that interval. If Janet has to wait more than 15 minutes for Donald, she gets bored and leaves. Donald is busier so will only wait 5 minutes for Janet. What is the probability that the two will eat together? Express your answer as a fraction.

Find all functions $f: \mathbb{R}^+\to\mathbb{R}^+$ such that \[f(x+f(y+xy))=(y+1)f(x+1)-1\]for all $x,y\in\mathbb{R}^+$. ($\mathbb{R}^+$ denotes the set of positive real numbers.)

7. Peggy picks three positive integers between $1$ and $25$, inclusive, and tells us the following information about those numbers: [list] [*] Exactly one of them is a multiple of $2$; [*] Exactly one of them is a multiple of $3$; [*] Exactly one of them is a multiple of $5$; [*] Exactly one of them is a multiple of $7$; [*] Exactly one of them is a multiple of $11$. [/list] What is the maximum possible sum of the integers that Peggy picked? 8. What is the largest positive integer $k$ such that $2^k$ divides $2^{4^8}+8^{2^4}+4^{8^2}$? 9. Find the smallest integer $n$ such that $n$ is the sum of $7$ consecutive positive integers and the sum of $12$ consecutive positive integers.

Carissa is crossing a very, very, very wide street, and did not properly check both ways before doing so. (Don't be like Carissa!) She initially begins walking at $2$ feet per second. Suddenly, she hears a car approaching, and begins running, eventually making it safely to the other side, half a minute after she began crossing. Given that Carissa always runs $n$ times as fast as she walks and that she spent $n$ times as much time running as she did walking, and given that the street is $260$ feet wide, find Carissa's running speed, in feet per second. [i]Proposed by Andrew Wu[/i]

Your team receives up to $100$ points total for the team round. To play this minigame for up to $10$ bonus points, you must decide how to construct an optimal army with number of soldiers equal to the points you receive. Construct an army of $100$ soldiers with $5$ flanks; thus your army is the union of battalions $B_1$, $B_2$, $B_3$, $B_4$, and $B_5$. You choose the size of each battalion such that $|B_1|+|B_2|+|B_3|+|B_4|+|B_5|=100$. The size of each batallion must be integral and non-negative. Then, suppose you receive $n$ points for the Team Round. We will then "supply" your army as follows: if $n>B_1$, we fill in battalion $1$ so that it has $|B_1|$ soldiers; then repeat for the next battalion with $n-|B_1|$ soldiers. If at some point there are not enough soldiers to fill the battalion, the remainder will be put in that battalion and subsequent battalions will be empty. (Ex: suppose you tell us to form battalions of size $\{20,30,20,20,10\}$, and your team scores $73$ points. Then your battalions will actually be $\{20,30,20,3,0\}$.) Your team's army will then "fight" another's. The $B_i$ of both teams will be compared with the other $B_i$, and the winner of the overall war is the army who wins the majority of the battalion fights. The winner receives $1$ victory point, and in case of ties, both teams receive $\tfrac12$ victory points. Every team's army will fight everyone else's and the team war score will be the sum of the victory points won from wars. The teams with ranking $x$ where $7k\leq x\leq 7(k+1)$ will earn $10-k$ bonus points. For example: Team Princeton decides to allocate its army into battalions with size $|B_1|$, $|B_2|$, $|B_3|$, $|B_4|$, $|B_5|$ $=$ $20$, $20$, $20$, $20$, $20$. Team MIT allocates its army into battalions with size $|B_1|$, $|B_2|$, $|B_3|$, $|B_4|$, $|B_5|$ $=$ $10$, $10$, $10$, $10$, $60$. Now suppose Princeton scores $80$ points on the Team Round, and MIT scores $90$ points. Then after supplying, the armies will actually look like $\{20, 20, 20, 20, 0\}$ for Princeton and $\{10, 10, 10, 10, 50\}$ for MIT. Then note that in a war, Princeton beats MIT in the first four battalion battles while MIT only wins the last battalion battle; therefore Princeton wins the war, and Princeton would win $1$ victory point.