The Dinky is a train connecting Princeton to the outside world. It runs on an odd schedule: the train arrive once every one-hour block at some uniformly random time (once at a random time between $\text{9am}$ and $\text{10am}$, once at a random time between $\text{10am}$ and $\text{11am}$, and so on). One day, Emilia arrives at the station, at some uniformly random time, and does not know the time. She expects to wait for $y$ minutes for the next train to arrive. After waiting for an hour, a train has still not come. She now expects to wait for $z$ minutes. Find $yz$.

The Princeton University Band plays a setlist of 8 distinct songs, 3 of which are tiring to play. If the Band can't play any two tiring songs in a row, how many ways can the band play its 8 songs?

Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price $p$. A day later, the U-Store decides to give a discount on plushies and sell them at $p'$ with $0 < p' < p$. Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has $12$ plushies, Bob has $40$, and Charlie has $52$ but they all spent the same amount of money: $\$42$. How many plushies did Alice buy on the first day?

Let $a, b, c, d$ three strictly positive real numbers such that \[a^{2}+b^{2}+c^{2}=d^{2}+e^{2},\] \[a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.\] Compare \[a^{3}+b^{3}+c^{3}\] with \[d^{3}+e^{3},\]

Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price $p$. A day later, the U-Store decides to give a discount on plushies and sell them at $p'$ with $0 < p' < p$. Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has $12$ plushies, Bob has $40$, and Charlie has $52$ but they all spent the same amount of money: $\$42$. How many plushies did Alice buy on the first day?

Let $\mathbb{N} = \{1, 2, 3, \dots\}$ be the set of all positive integers, and let $f$ be a bijection from $\mathbb{N}$ to $\mathbb{N}$. Must there exist some positive integer $n$ such that $(f(1), f(2), \dots, f(n))$ is a permutation of $(1, 2, \dots, n)$?

You have a triangle, $ABC$. Draw in the internal angle trisectors. Let the two trisectors closest to $AB$ intersect at $D$, the two trisectors closest to $BC$ intersect at $E$, and the two closest to $AC$ at $F$. Prove that $DEF$ is equilateral.

Including the original, how many ways are there to rearrange the letters in PRINCETON so that no two vowels (I, E, O) are consecutive and no three consonants (P, R, N, C, T, N) are consecutive?

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.