## Catharsis as 2005 USAMO Problem Set

*After Lucy Zhang*

**Problem 1**

**( Liwei Lu) Suppose that I have n people in a room, some who have betrayed each other, and some who have not. Assume that some betrayals are two-sided (i.e. both parties have betrayed each other and have full knowledge of the other’s betrayal) and that some are one-sided (one party does not know that the other has betrayed them). Given that n is composite, find the ratio of one-sided betrayals to two-sided betrayals. **

Note that it suffices to prove the innate selfishness of human nature in order to prove that the ratio of one-sided betrayals to two-sided betrayals is undefined. Some mathematicians may denote this as infinite, but for the purposes of this solution, we will call it undefined. Other solutions may find it useful to employ the handshaking lemma.

He betrayed me in August first by promising that he would wait two years for me. I thought we had established a Nash equilibrium where both of us would stay with each other for the long haul even though we were 3,000 miles apart. What I didn’t realize is that I assumed he had nothing to gain by switching his strategy.

Liwei called me a few days ago, but I realized that I didn’t have anything to say to him. So I smiled, cleared my throat, and told him to go to hell.

**Problem 2**

**( Lillian Temple) Write the general form of all quadratic systems with real coefficients such that there do not exist any real or complex solutions.**

In quadratic systems, it’s always important to remember that the square can result in imaginary solutions. In this case, the question asks for no real or complex solutions, which simplifies our work.

Simply dream about a pretty college girl in California somewhere. Flatten her into a Cartesian coordinate system and trace out her curves onto the paper. Approximate her hips into parabolas and her lips into ellipses. Once you’re done, tear up the paper and toss it. You have your solution: she is not real or complex: she is the general form of your desired quadratic system.

**Problem 3**

**( Audrey Kuang) Construct a cyclic quadrilateral ABCD such that ABCD lies on a circle of radius 13. Let P and Q be two points on sides AB and CD, respectively. Find all integer values of PQ such that PQ will never intersect with AC and BD. **

I was disgustingly infatuated with a girl from my Chinese school. I had skipped a year of school over the summer just so I could be in the same grade as her. In our last year, we both took AP Chinese (although I never took the exam because I was scared to fail). She took the exam and she got a 5. I never did much in that class, only sat around fiddling with my pencil, carefully watching my classmates’ tongues form words that I could never understand. I tried to shape my mouth like theirs but it came out all clumsy, like a baby bird trying to fly for the first time.

Audrey Kuang was the first girl I ever loved, and the first girl I ever hated. I never hurt her, never punched her in the teeth or called her names, but I hated her to death. She and I were exactly the same: both Chinese church girls that were on the fast track to suicide and Ivies. I think I hated her because she was so good at stitching up a facade, while my needles kept poking me under the fingernails no matter how many thimbles I wore. She earned the nickname of weaver girl, and I would have done anything to be the cowherd, except that I remembered that they only met once a year. Would it be better to feel everything on one day a year, or to feel some pale imitation of love on all the other days of the year?

We never talked in that class even though we sat right next to each other, even though we resided in the same classroom for two hours every Saturday. I thought that if I reached across the desk and touched her fingers, something would click.

I found her on Instagram a few weeks ago. I was surprised by how very little she had changed: she still wore her hair in two braids that wept down the sides of her shoulders, still wore dresses that were too tight in the waist. I was pleasantly shocked to discover that she still liked to weave together tapestries of pretense. I call myself cowherd then, coax her name into my mouth and clumsily circle the syllables once more.

**Problem 4**

**( Harry Takahashi) A kitchen table is twenty-three years old and has four legs each of length L, each forming a 23º angle with the floor. Assume that every day, the Yang family places one object on it. This can include anything from the very mundane (Dad’s glasses, Christine’s headphones) to the very special (Mom’s disappointment after I didn’t get into Harvard like my sister, Justin’s fetish for younger girls). If every object takes up finite space but has infinite worth, prove that the infinite summation of the objects will never cause the table to collapse.**

It feels like time has been with me up to my 16th birthday, and I can’t ever catch a breath. I don’t care for a sweet sixteen and college applications don’t feel real until you’re staring at twenty-four open tabs desperately wishing for time to be on your side. I don’t even have to work on these essays yet. I’m just scared that nowhere will accept me.

I trace the scratches on the kitchen table and feel my fingernail scrape against the big scratch made when my father dropped his computer after thirteen sleepless nights. I remember my mother told him to stop working, and even though it was already three in the morning, she made me get in the car and drive all the way up to Flushing just to go to 敦煌兰州牛肉面 and get my father’s favorite noodles. I technically wasn’t allowed to drive alone yet, so I stuck to the back roads and stayed below 60 miles per hour the whole way.

By the time I got home, the noodles were already soggy but that didn’t matter to my mother. She cleared away his papers and filed them under “H” for “He works too hard.” The kitchen table angled its legs again to better bear the new weight of the noodle bowl. We took everything else off the table, and we waited for the sky to rattle itself powder blue again.

**Problem 5**

**( Catherine Jiang) Let K be represented by a bipartite graph whose vertices fall into one of two disjoint sets: harmony and dissonance. The chromatic number is the smallest number of colors needed to color every vertex such that no two adjacent vertices are the same color. A balancing line is a line that when drawn, will have an equal number of “harmony” and “dissonance” vertices on either side. Show that all friendships are a subset of K. **

Nothing was strange about our friendship. We fought and tore each other’s hair out and clawed our nails down each other’s arms like any other friendship. I dreaded having to see her every day. Still, though, we shared secrets like no other best friends did. I told her when I first got my period and she told me that getting my period meant I was a woman now. We looked down at our formless legs and our band camp t-shirts and claimed womanhood to be ours for the taking.

When high school rolled around she changed her name to Catherine Johnson and skipped off to hang out with the kids who vaped behind the school building every day before classes. I was confused. How did she just shed her Chinese like a snakeskin? Still, I said, okay, but we’ll still walk home after school, right? I didn’t hear her reply but I assumed she said yes.

I waited three hours for her after school. I missed my bus and two friends’ invites to drive me home just for her. I called her up later that night and asked where she went. She said, “Oh my God, you won’t *believe* where they drove me to!” I pretended to listen to her adventures at the Catholic boarding school where they had graffitied the walls with female genitalia. I said I had to go because I had calculus homework. “Boo, I miss you so much,” she moaned over the phone, but I noticed she hung up first.

We don’t eat lunch together anymore, and I still haven’t told her that I lost my virginity in Audrey’s bedroom. It didn’t bleed or hurt, even though Catherine said it would. I told Catherine I liked her yellow pleated skirt yesterday. She said thanks. We never have another conversation again. I’m good with that.

**Problem 6**

**( Julia Kim) Verify the following tautologies, assuming that every sentence is a nonempty set where every element (word) can be represented by a broken fantasy. Assume standard tautology notation: ⇒ means implies, v means and, ^ means or, ~ means the negation of, and this entire set of tautologies means I have finally let you go. **

**(If you loved me once ⇒ You can love me again) v (If you love me again ⇒ Why did you ever stop?)**

**~(We aren’t on the same journey anymore) ^ (I never stopped remembering you) **

**((I once knew everything about you ⇒ Too much time has passed between us) ^ (Too much time has passed between us ⇒ You are now entirely unfamiliar)) ⇒ (I once knew everything about you ⇒ You are now entirely unfamiliar)**

This proof has been left as an exercise to the reader.

Alright, Julia. I’m done.

Senna Xiang is a teen writer. Her work is published in *Peach Magazine* and *Superfroot Magazine*, among other lovely places.