Given two positive integers satisfying and their sum is an even number, consider the quadratic polynomial:


Prove that all roots of are positive integers but are not perfect squares.

Prove that there exist infinitely many positive integers m such that the equation (xz+1)(yz+1) = mz^3 + 1 has infinitely many positive integer solutions.

Find the largest positive real number such that for any positive integer , satisfies .

Let be a positive integer. For every positive integer the sequence is defined, where are integers. Among these sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by ?

Find all odd integers such that
for some positive integers and .

I'm turning 18 and realize how useless I am.

I'm a week away from forever leaving behind childhood. Though I perfectly understand that reaching the milestone of having lived 567,648,000 seconds carries mere symbolic meaning, I believe it serves as a stressful but necessary reminder for me to evaluate my life up to this point and prepare myself going forward. This has proven to be a nightmare.

My life right now doesn't make much sense. I know this is my old INTP Ti function at play, but how tf does one logically explain my behaviors lately:
-Drinking alcohol
-Reading on AI and pondering the future of humanity
-Spending hours on Quora reading different perspectives on a same topic before moving on
-Weightlifting/exercising like mad
-Taking boxing lessons
-Killing my gains by periods of binge-eating
-Running around the city at night like a bad boy who dgaf
-Mass e-mailing Stanford CS/math professors
-Reading about 18th century and 19th century philosophers
-Learning data science from online tutorials
-Going to clubs
-Rapping and dancing around my house at night.
-Spending a ton of time applying to summer programs I don't want to go to (besides SPARC <3) and secretly hoping I get rejected so I don't have to go
-Listening to lectures given by Silicon Valley venture capitalists and trying to understand their thought process
-Getting addicted to playing chess with my close friend at school
-Writing raps (this is actually really fun)

My dreams, which used to be a source of inspiration, are now:
-Full of inappropriate sh*t
This likely means my dreams which ran on inspirationare now running on fumes.
-Simulations of random events which I'm a part of that're so eerily realistic I wonder if they'll happen in the future
This likely means the present is so insignificant that my mind is on autopilot exploring the future. This also means my low confidence in predicting the future.
-Conversations with acquaintances that I sometimes confuse with ones that actually happened
This means I'm probably lonely and desire human interaction.

After the difficult process of analyzing my actions and categorizing them under common themes, I believe my actions are motivated by two states of mind:
1. I'm fed up at how uncertain about the future I am.
2. I feel lonely and want to be cared for.

As a response to the first, I try to add structure to my life: reading/writing, setting tough new goals to keep busy and thinking/planning the future even when it's inefficient to do so so early (people I've talked to told me planning my life at 17 isn't very effective).

As a response to the second, I try to increase my objective attractiveness, get closer around people so I can observe them, and seek attention. I try to participate more in school activities, despite not being able to connect to most people at school. This actually makes me more tired and sadder since it's all superficial. I realized it's actually very easy for me to feel happy. Last week, I was in Japan, and I realized the happiest I felt during the trip was actually the last night, when my relatives and I played a few hours of poker (I'm actually so bad I ended up last). I also feel happy playing chess with my friend (I also usually lose; contrary to popular belief I'm not smart). In fact, just having someone who listens and tries to understand me makes me very, very, very happy. But because I'm Mr. Stanford and Mr. You're-My-[Kid's]-Role-Model, The Legend, the people who accept me for who I am are extremely low in supply (which is also why I place high value on the few who do).

Emotions have also had a spillover effect lately. Though I have systems in place that prevent emotions (which takes a lot of energy to analyze so I usually don't) from affecting my productivity and decision-making, I realize the maintenance cost of storing my emotions in a box are steadily increasing - mainly because the constant need to upgrade and buy larger boxes. Perhaps I took up rapping/writing forms of self-expression as ways to relieve some of my emotions, though this often isn't reliable and hard to maintain.

The thing is, my actions taken in response to state of mind #1 takes away from time for socialization, and my inaptness at planning the future makes me feel more vulnerable. This reinforces state of mind #2, which reinforces a feeling of worthlessness and uncertainty. This creates a loop that I find extremely hard to get out of.

Then again, I've fell into these crises before, but have always managed to delay them for later. I for some reason don't find it as urgent of a priority to get out of this loop and reinvigorate my productivity right now, simply because for the first time after years of hustling I can rest for a while. Taking advice from friend spartan168 at Stanford, it's probably wise to rest while I still can. Sadly, I simply don't know how to "enjoy" life.

To elaborate on the last point, I was so pissed when I had to go to vacation in Japan when I could've travelled to two cities in China to continue my job of guest speaking for my job. Instead, I had to walk around dumb gardens and take pictures under Sakura trees when I could've been inspiring hundreds of parents and students. This makes me feel fcking selfish. It also makes me realize enjoying life is an inverse-relation activity to my purpose in life - to improve the world and the people in it. Occasionally, I miss the days when I did math into ungodly hours at night - it was when I enjoyed doing something for myself while working towards my goals. Nowadays, I feel like my life is the sum of my "impact" in the world, the present meager amount of which makes me feel sadder.

In conclusion, by writing this post, I've identified my two main needs in life - (1) be in a position to create more and more meaningful impact and (2) people who accept me for who I am. Even though I try to do always what's most logical, I will keep these two fundamental values in mind for the rest of my [adult] life.

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I will make a final and last post on this blog next week when I turn 18 with some parting advice/notes/contact-info. The blog will be locked a few days after. This is my last life update on here folks.

Firstly, the post I alluded to on my new site is complete. Reads, subscriptions, bookmarks and/or comments are appreciated.

Much of my life, especially recent years, I've struggled with the expression of emotions. This is different from understanding emotions and people, which I think I do pretty well (I have yet to mis-guess the mbti of people I know in real life, though I did stumble over the mbti's of some facebook friends). When I asked a friend on what she thought emotions were for, she responded: gxah, I'll try my best to preserve your anonymity.

Emotions certainly have helped bridge differences between humans. Body language actually accounts for the majority of in-person communication, not vocal. While it has come a long way, we should recognize emotions first formed in our limbic systems as fear, excitement, love, in our mammalian ancestors as tools for the survival and development. After all, fear is incentive to stay alive, excitement to amass resources, love to reproduce, etc. Over centuries, however, along the origin of our species we developed the unique neocortex for primates to choose rational thinking for our decision making process. Yet, we still relied on the limbic system due to its benefit of lower mental energy consumption and instant gratification. In our society today, such motivations still manifest in the form of drugs, sex, pornography, etc. which continues to lead people astray.

This is not to say I believe emotions should be outright eradicated (though it might be possible with brain-machine interfaces), but that we should understand how we deal with emotions and express it in appropriate ways, something I now understand after years of being misguided.

Many years ago, I really, really liked this girl. But she told me (many times) she didn't feel the same way. Today, I would immediately redraw my investment after the first such response, but being the hard-headed boy I was back then I dwelled in my emotions and continued on. By my current value system, I basically became dysfunctional, and (over) thought how I could get her to requite my feelings. I exaggerated the implied meanings of meaningless actions of hers to persuade myself I had a chance (cognitive bias), so it didn't help when my friends sort of entertained themselves watching me rather than straight up telling me to give up.

The reason why I believe I was so hard-headed, other than I am hard-headed (still) in everything I want to achieve, is the failures of my previous crushes. I got my first crush in 5th grade, and her being 100% extrovert and me 100% introvert, never found the chance to approach her. She left me hanging on a cliff when she left after the year. My second crush, also in 5th grade, rejected me in all-caps with explicit language on Skype and blocked me after finding out I like her, the worst part being I was bad at English and couldn't understand half her swear words and sarcasm (searching it up afterwards was more painful). Having been treated like nothing for years at that point, these failures only fed my dogged determination to prove others wrong, and is probably why shiningsunnyday the avenger was so hard-headed for this girl.

Granted, this was unfair for her, the vessel for my journey to expressing my identity, but she helped me grow in ways I wouldn't have otherwise. In fact, spending sleepless nights on threads on "how to get a girl to like you" developed my EQ, humor, and social skills a lot, and is why I'm interested and plan to study more psychology in college.

This would've been a terrific self-development journey, bar the one devil in the details: most of our communication took place online. This led to stagnate my ability to express emotions.

As one of my biggest weaknesses, expression of emotions is tough not because of this girl, but also because my time growing up. Having my parents divorced in third grade (and also being the only child) never allowed me to be in the influence of genuine expressions of emotions, since my parents would "act" whenever I'm around, and it certainly doesn't help that they're both terrible actors.

Though I may seem stoned time-to-time, this doesn't mean I don't feel emotions. In fact, I probably do more than anyone, because emotions that're meant to be expressed remain bottled up in me often. I take pride in my ability to "fake" my behavior to act like anyone (since years of being alone in elementary school allowed me to observe and study all types of social behavior). As a defensive measure, my solution as of now is to catch my emotions, make sure it doesn't get in the way of productivity, and dilute it in different ways, like writing blog posts or rapping on stage to the cheers of a crowd. So all in all, I feel really comfortable with my mechanisms of dealing with emotions. It's actually an amazing lifestyle to have, not having to worry about emotions getting in the way - like cruising through college with a scholarship as opposed to loans. I can also choose to trap and activate emotions, just to analyze them or for fun.

I also find myself liking girls as frequently as any normal teen, which feels natural as long as I don't fall for one, which I've yet to experience ever since the subject "girl" of this post. I find myself liking many girls, often those who possess an non-empty subset of the characteristics I desire in a partner.

I do worry of the possibility of being hacked, however - if a girl ever gets me in an emotional chokehold, aka love. Love, unlike "like", is unfamiliar waters for me (my most recent experience being the "girl"), so I plan to stay away from it as best as I can, until after marriage. If it does come up, I am confident in my immune system to get rid of the parasite before it begins its damage.

Hopefully, you found this post intriguing and applicable.

I used to be a believer in parallel universes rather than simulation theory. After all, what's there to stop me from stopping my typing at the end of this sentence and hit the club to chug down some shots and do god-knows what? As a proud INTP for the majority of my life, introverted thinking (Ti) drove me to constantly intake new information and imagine the what if. In fact, I almost prided myself as being an INTP in my college apps, as my response to one college prompt went something like this:

Growing up, the internet become my escape. I would be reading SpaceX's website one moment and the next be watching a video explaining relativity in simple terms. My childhood naivete for seeking knowledge was my escape from everyday life, amongst Naruto, Skyrim, and wealth of anime and video games.
Gradually, my interest dampened on these things. The best way to put it is, sh*t just became boringly predictable. I was a class clown in middle school, so I've always had this knack of knowing how my actions would affect others' impressions of me. Pull a string here during a 6th grade english class impersonation, and the girls would laugh. Make an innuendo there, and the next moment a frown would be on the teacher's face. I always wanted attention, so most of socialization came in the form of trying different things, seeing how people respond, and adjusting my actions. I thought I could get anything I wanted in life, and those parent college talks was the stuff of the hour, so I thought, cool, I'll figure out how to get into a good college. Truth be told, who I am today was nearly the same person I envisioned myself to be 4 years ago, when I walked out of my middle school homeroom for the last time, held up a finger at the end of the hall (it was late at night), and shouted out, "I will succeed!"

Today, my world view are mostly the same, except even the once-blurry details are now all the more obvious. I no longer expend the energy to entertain the what-if, for I'm busy figuring out the what-is and the what-will. Most of modern society life to me seems now like a particle undergoing motion while defined by an underlying set of rules of physics. These real life occurrences appear to be the result of probability but their causes themselves are the result of actions that came before then, hence the saying "Everything happens for a reason." Sure, the rate at which possibilities multiply is near infinite, hence the illusion that everything seems possible. But really, once these underlying set of rules are known, most occurrences are predictable. Like in a video game, the NPCs seem to move about randomly, but they're the result of pseudo-random algorithms that are random enough to make us think they're random, but in reality are not. This is now why I believe in simulation theory.

If you think about it, most of our behaviors are becoming more and more predictable. The more competitive society becomes, the more our behavior is driven by the need to optimize our standing in society, because there's only a finite amount of resources (power, fame, money). When we like someone else's totally-undeserving pic of a random cake or something, it's not because we genuinely appraised it to be worthy of it, but because subconsciously our brain wires it to a higher chance the said someone will like our future pics, which is wired to a likelihood of higher standing. It's an uncomfortable truth, but most of our actions are explained by psychology. Being halfway done with the book Influence: Science and Practice, I learned a lot of ways salespeople and people in general manipulate in others using an understanding of basic psychological and civilizational principles, and it's astounding how all of this comes together. Essentially, we've become AI to the point everyone wants to optimize their actions to achieve a certain outcome most consistent with their values/belief (could be for money, power, being liked, or combination of any means of an end). Experience fine tunes our behavior model for optimization (like evolution but sped up so we can constantly see the updated results). It doesn't make sense that we whine and say AI will take away the spirits that make us human - curiosity, genuineness, passion, etc. - because honestly, we're already becoming AI ourselves (it's driven by evolutionary mechanisms that we must act in certain ways for us to survive in society). When I meet some mutual family friends, their children (who undergo the Chinese GaoKao), have essentially become robots. "6 AM wake up every morning. 7-10 AM language tutoring. 10-11 AM tennis practice. 11-12 lunch at cafeteria. 12-1 TOEFL tutoring. 1-3 math and science tutoring, etc." I had to pretend to listen intently as their parents tell me their children's academic regime. Then they give me that surely-you-understand-this-for-you-went-through-it-yourself-and-now-you're-on-top look. "What advice can you give?" I usually just provide some general template and act polite. I do understand, and it's not that I don't want to help, but more that I've seen through the system itself completely and it doesn't work, less so the further ahead I look. In 10 years, majority of education will be personalized online. In 20 years, teachers will be replaced by AI. In 30 years, brain implants will be available that provide the most direct form of education. By midcentury, our brains would be connected to a cloud. In this future where googling for information is automated inside our brain on a subconscious level, it's not just fitting to say what we're doing now is meaningless, it's akin to serving a sentence in jail. Of course, those in my family I tell this to think I'm ridiculous, but it will happen (and I will be a part of it). But the bigger question at hand is, what are the things that are still meaningful, now, in 10 years, 20 years, forever? How can we ingrain purpose for humans when AI takes over society and UBI introduced? What can be preserved and what must go, and how will we adapt? I toss and turn a lot of such schemes in my head. That explains why I'm now an INTJ.

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So no, I didn't take the AMC and didn't plan to take any math contests this year. There're bigger problems that we must solve and their links to the fate of humanity are ones that only our generation can create and secure.

In real life, I roam around a lot now to all sorts of places in the city (almost none of my energy is expended on day-to-day life, so I guess that makes me in the eyes of others spontaneous af). Sigh, I wish I was less lonely. An ENFP/ENFJ girlfriend (or any intuitive-mbti-types) would be nice company until college starts and provide me of some last pre-adulthood fun, at which point its time to start turning these crazy schemes of mine into reality (funding, VCs, business plans, learning markets, startups...). As graduation approaches, I guess I'll keep making my plans.

Check out my website for more elaboration of the visions/schemes I have in mind in the near and distant future.

Points and lie respectively on sides and of a triangle and . Segments and cross at . Circumscribed circles of triangles and cross again at point different from . Prove that point lies on the bisector of angle .

Let be a triangle with circum-circle . Let be a point in the interior of triangle which is also on the bisector of . Let meet in respectively. Suppose is the point of intersection of with ; and is the point of intersection of with . Prove that is parallel to .