Just for some context, the WWP^2 ARML Team Selection Test is used to determine the two teams that will be sent to the Penn State University, where the team is to compete at the American Regions Mathematics League (ARML). Students are selected from three schools: namely WWP High School North, WWP High School South and Princeton High School. It is extremely selective; the top fifteen students are selected for the A team and the next fifteen students are selected for the B team.
April 30th, 2022
I arrived at the Plainsboro Public Library at about 2:40 PM, where we were to take the test.
We were to select a one-hour block between 2 to 4:30 PM. Some of my friends have already arrived and were taking the test when I arrived.
Not knowing what to do, I promptly questioned a lady at the counter where to go for the “math tryout test”. She directed me to the Cafe Area, which was an open space with a couple of couches, tables, and chairs that looked more like a lounge rather than a snacking zone.
I sat on a couch and scrolled through Discord to ask one of the officers if I was supposed to be there, as I didn’t see any familiar math club people that were waiting at my location. In the meantime, I spent some time reviewing some random formulas that I could recall (shoelace theorem – first coordinate at the ends!). I recalled how much I needed to rely on them, but it was equally important to think about the correct insight to solve the problem.
After about ten minutes staring out of the tall glass windows, Aadarsh arrived at the lobby. Since he looked quite confused, I gestured to him to come and told him to wait.
Amusingly enough, Edward (one of the officers) suddenly came up to us and waved a couple of test papers in front of our faces. I noticed that he was wearing an ARML shirt that he had presumably acquired from one of the past competitions.
He asked us why we were waiting and we explained. He shrugged and directed us to a table of medium length. Aadarsh sat on one end and I sat on the diagonally opposite end. After I got my scratch paper and pencils, Aathreya (another officer and a close friend) came to me and gave me the test paper. We greeted each other casually as he wrote the start time on one corner. I secretly regretted my existence as I wrote my AMC 10 score (94.5) below my name.
And so the test begun. I flipped the sheet over and worked on the problems with my full concentration. A couple of (maybe college?) girls were discussing their irrelevant gossip at the table next to us, but I did not heed them and proceeded to work.
Here’s a problem-by-problem breakdown: (I may have messed up the numbering or something so I’ll fix any errors once I get a hold of the TST paper)
Problem 1: This problem was just asking for the number of positive divisors of 96. You just prime factorize and you’re done. Easy peasy lemon squeezy.
Problem 2: I may have messed this one up actually. Although it was a problem involving trianglular numbers (which I solved quite quickly), I forgot whether or not to count the original number. Oh well…
Problem 3: Find the area of an octagon with side length 2. Just use complementary area (a big square minus four litle triangles) and a bunch of 45-45-90 work. Very easy.
Amazingly enough, Sid Basu memorized the formula for this. But it only saves you about a minute…
Problem 4: Another easy problem: just the system of equations x-y=11 and xy=6. Use substitution to find the two possible values of x (uhh how is this competition math level?)
Problem 5: Oh my, this was actually very easy too. Everything collapses when you take mod 5. This reduces to p,p+1,p+2,p+3, and p+4 mod 5. This means that no matter what value of p you choose (other than 5), at least one of the numbers will be divisible by 5, making the statement untrue. Hence our only prime p that works is p=5.
Problem 6: This is the point where things get very tricky. I put 55 mostly by guessing and no mathematical proof (1+2+..+10), but yeah I’m not too good at expected value problems.
Problem 7: Why, oh why does this problem exist. I spent almost twenty minutes PIE BASHING with casework and didn’t get anywhere. After the test Romir suggested recursion as an approach, which I think it might work.
Problem 8: I’m pretty sure this one was the cyclic quadrilateral one. Look, I was about to put 28 but I knew that the diagonals weren’t limited to integer side lengths. I proceeded to use similar triangles and bashed it out the wrong way..
Basically, you use Ptolemy’s theorem to get (AM)(RL) = 195. Then, you notice that one of the diagonals is a diameter of the circle because 5^2+14^2 = 10^2+11^2 and the result follows from the converse of the Pythagorean Theorem. The rest is easy. AGH why didn’t I think of this during the contest.
Problem 9: What a troll problem. THANKS EDWARD!
You use difference of squares repeatidly and see that some of the products don’t have real roots. Then after finding a pattern you get that f^2022(x) must have 2023 real roots. I put 4043 during the test since I was being dumb.
Problem 10-11: I didn’t spend any time on these problems. However, Sid told me that he attempted the spheres problem (#11 I think) and he got several cross sections. Yeah, 3D geo is just 2D geo on drugs.
Problem 12 (last problem): This reminded me of inversions and sorting algorithms in computer science. (in a sequence a_1, .. a_n, an inversion is defined as two terms a_i, a_j such that a_i>a_j but i<j)
The problem basically asks for the expected number of inversions. Let’s just say I really messed this problem up. THANKS EDWARD!
During the test I continually switched my focus around with these problems (as you should in AMC, but not in the AIME or USAMO). Once the sixty minutes was up, Edward collected my paper. Over in the lounge area, I saw Suhas, AVD, Romir and other kids from the other two schools taking the exam. I still didn’t see Sid anywhere (later I found out that he left at around 3). I waited for my parents to pick me up and during that time, I wished my friends good luck.
I left the library thinking about the problems and regretting my bash for the cyclic quad problem.
With hope, I entered the car, hoping for my ARML team qualification.
Two days ago, I recieved an email that confirmed my hopes. In a month, I will be participating in the competition.