mop first half

by OronSH, Jun 15, 2024, 3:20 PM

Mop has been a lot of fun so far! I will try to include everything that I remember.

Classes
Classes have been pretty interesting so far (I'm in green mop). Unfortunately, only one class has been on geo

(we have had like 3 classes on intpoly)
In class, I usually sit in the back with popop614, cj13609517288, vsamc (and they usually groupsolve some 50 mohs problem while i struggle to understand them rip)
Here are some interesting problems from various classes.

Hallucination wrote:

Let $a_1,\dots,a_n$ be integers satisfying $i-n\le a_i\le i-1.$ Show there exists a nonempty subset $S$ of $\{1,2,\dots,n\}$ satisfying $\sum_{s\in S}a_i=0.$

Pell's Equation wrote:

Prove that there exist infinitely many positive integers $n$ such that $n^2+1$ divides $n!.$

Functional equations wrote:

Find all continuous functions $f:\Bbb R\to\Bbb R$ with $f(xy)=xf(y)+yf(x).$

Cauchy-Schwarz wrote:

Show that for any $x,y,z>1:\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}\le\sqrt{xyz+x}.$

Cubic Discriminant wrote:

Let $P$ be a monic polynomial of degree $d$ with integer coefficients. Suppose that all of its roots are real, distinct, and do not exceed $M$ in absolute value. Prove that for any of it roots $a,b$ the inequality $|a-b|\ge(2M)^{1-d}$ holds.

There have also been some interesting lectures on various topics: one on algebraic curves (or something) and a theorem with $4$ exceptions, one on tetrahedra with rational dihedral angles and a result with $59$ exceptions (at least two of which contain the substring 1434), and one on SAT solvers, which basically nukes all combo problems.

Tests
We have had $5$ tests so far. Can't say much about them yet

These tests have not been serious (e.g. the "buh" countdown to the start of each test). Unfortunately, the ELMO day 2 today is the last "non-serious" test (TSTST soon :wacko:

)
Also, participated in ISL guessing (the test problems are from the IMO shortlist, and you have to guess what position they are) and got second place in one test under the name "skibidi kibidi ibidi bidi idi di i"

Activities
I have played a lot of games so far at mop, I will try to list/explain every game I remember:
- Set: just regular set, trying to match 3 cards that are either all the same or all different on each of 4 properties.
- Continuous Set: like set, but instead of having 12 cards placed down, cards are continuously added to the board.
- Memory Set: cards are placed face down. In a turn, you can flip over 3 cards. If they are a set, you take them out, otherwise you flip them back over. (This is really hard and best played collaboratively)
- Set Uno: players start with 7 set cards and go around in turns, on your turn you can place a card in the pile that must have at least 2 of the 4 properties matching to the top card in the pile. If you don't have any working cards, draw cards until you get one. If the card you play forms a set with the top 2 cards in the pile, next player gets +3 cards and skips their turn. First player to run out of cards wins.

- Bomb party: you try to form a word containing a given string before your time runs out. If you use all the letters in the alphabet (except X and Z) you get an extra life. popop614 is very good at this game and has some crazy plays, once we managed to have a single game last for an hour!
- Verbal bomb party: you name a three letter string and everyone else has to come up with a word containing that string. This can be much harder than regular bomb party. This is fun to play at dinner.

- Crazy pool: played with the cue ball and 8 ball, players have to take turns rolling the cue ball at the 8 ball (with their hands). If the 8 ball stops moving on some player's turn, they lose a life. If some player sinks the 8 ball into one of the pockets, the next player loses a life.
- 4 pool: basically regular pool but the 4 ball is the cue ball. Since the 4 ball is solid, this means that whichever team has the solid ball never scratches, and whichever team has the stripes always scratches (since the first collision is always with a solid ball). It gets hard towards the end for solid team, since after sinking all the other solid balls, they have to first sink the 4 ball (which is really easy since the previous hit must be a scratch), but then they have one shot at hitting the 8 ball into a pocket using the cue ball. If they miss, it's a scratch and the 4 ball has to come back out.
- Around the world: basically, you are always starting at one corner of the table, and you are trying to roll it (consecutively) into every other hole. This is really hard since on the later holes you need some crazy bounces.

Here are some other things I've done:
- Went to Waffallonia (a waffle place) on Sunday with like basically everyone else in mop.
- Groupsolved p83 from qwerty123456asdfgzxcvb's geo book at 1am (it was really nice).
- Played frisbee (or as I called it, "fribidi") with a bunch of people. It turns out, I'm probably the worst mopper at frisbee. :showoff:

- Last night, performed a beautiful demonstration (read: skibidi) of the profound fact that $3\mid 2^6-1$ with popop614, polishedhardwoodtable, GrantStar, vsamc, john0512. We also did various other demonstrations of other mathematical insights.

I've pretty much forgotten everything else. :oops_sign: