Cn/lnn bound for S

by EthanWYX2009, Mar 18, 2025, 8:24 AM

Prove that there exists an constant $C,$ such that for all integer $n\ge 2$ and a subset $S$ of $[n],$ satisfy $a\mid\tbinom ab$ for all $a,b\in S,$ $a>b,$ then $|S|\le \frac{Cn}{\ln n}.$

Created by Yuxing Ye

number theory

Polygon formed by the edges of an infinite chessboard

by AlperenINAN, Mar 18, 2025, 6:27 AM

Let $P$ be a polygon formed by the edges of an infinite chessboard, which does not intersect itself. Let the numbers $a_1,a_2,a_3$ represent the number of unit squares that have exactly $1,2\text{ or } 3$ edges on the boundary of $P$ respectively. Find the largest real number $k$ such that the inequality $a_1+a_2>ka_3$ holds for each polygon constructed with these conditions.

combinatorics

Nice FE as the First Day Finale

by swynca, Mar 18, 2025, 6:24 AM

Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for all $x,y \in \mathbb{R}-\{0\}$ ,
$f(x) \neq 0 \text{ and } \frac{f(x)}{f(y)} + \frac{f(y)}{f(x)} - f \left( \frac{x}{y}-\frac{y}{x} \right) =2$

functional equation

Turkey

algebra

Interesting inequality

by sqing, Mar 18, 2025, 4:16 AM

Let $a,b\geq 2 .$ Prove that
$(a^2-1)(b^2-1) -6ab\geq-15$ $(a^2-1)(b^2-1) -7ab\geq -\frac{58}{3}$ $(a^3-1)(b^3-1) -\frac{21}{4}a^2b^2\geq -35$ $(a^3-1)(b^3-1) -6a^2b^2\geq-\frac{2391}{49}$

This post has been edited 2 times. Last edited by sqing, 5 hours ago

inequalities proposed

inequalities

Inspired by my own results

by sqing, Mar 17, 2025, 8:32 AM

Let $a , b$ be reals such that $a+b+ab=1.$ Show that $1-\frac{1 }{\sqrt2}\le \frac{1}{a^2+1}+\frac{1}{b^2+1}\le 1+\frac{1 }{\sqrt2}$ Let $a , b\geq 0$ and $a+b+ab=1.$ Show that $\frac{3}{2}\le \frac{1}{a^2+1}+\frac{1}{b^2+1}\le 1+\frac{1 }{\sqrt2}$

inequalities proposed

algebra

Problem 2830

by sqing, Mar 17, 2025, 8:15 AM

Let $a,b>0$ and $\frac{1}{a^2+1}+ \frac{1}{b^2+1}=t$ $(1<t<2).$ Find the value range of $a+b.$
h

This post has been edited 1 time. Last edited by sqing, Yesterday at 8:24 AM

inequalities proposed

algebra

hard problem

by Noname23, Mar 16, 2025, 4:57 PM

problem

This post has been edited 1 time. Last edited by Noname23, Sunday at 5:34 PM

inequalities

Natural function and cubelike expression

by sarjinius, Mar 9, 2025, 3:42 PM

Let $\mathbb{N}$ be the set of positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for all $m, n \in \mathbb{N}$ , $m^2f(m) + n^2f(n) + 3mn(m + n)$ is a perfect cube.

number theory

function

perfect cube

Roots, bounding and other delusions

by anantmudgal09, Mar 7, 2021, 10:33 AM

Let $\mathbb{R}[x]$ be the set of all polynomials with real coefficients. Find all functions $f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$ satisfying the following conditions:

$f$ maps the zero polynomial to itself,
for any non-zero polynomial $P \in \mathbb{R}[x]$ , $\text{deg} \, f(P) \le 1+ \text{deg} \, P$ , and
for any two polynomials $P, Q \in \mathbb{R}[x]$ , the polynomials $P-f(Q)$ and $Q-f(P)$ have the same set of real roots.

Proposed by Anant Mudgal, Sutanay Bhattacharya, Pulkit Sinha

This post has been edited 1 time. Last edited by anantmudgal09, Mar 7, 2021, 5:20 PM
Reason: I had to do this after Ankoganit suggested it. Missed opportunity.

square geometry bisect $\angle ESB$

by GorgonMathDota, Nov 8, 2020, 12:52 AM

Let $ABCD$ be a square of center $O$ and let $M$ be the symmetric of the point $B$ with respect to point $A$ . Let $E$ be the intersection of $CM$ and $BD$ , and let $S$ be the intersection of $MO$ and $AE$ . Show that $SO$ is the angle bisector of $\angle ESB$ .

This post has been edited 2 times. Last edited by GorgonMathDota, Nov 8, 2020, 1:10 AM

2018 Motivation

by shiningsunnyday, Jan 1, 2018, 12:06 PM

Last year's "2017 Motivation" post worked out pretty well, so here goes another one for this year. I'll save goals for last, and talk about what's been transpiring lately.

During the first week of winter break, I mostly stayed home at coded a lot in Python. It's way easier to understand and write than Java, and I did a lot of data science stuff with it on DataCamp, and am about two-thirds done with obtaining a career track certificate for a Python Programmer! (and about one-third from Data Scientist with Python).

I also began participating in a few CodeForces contests, but realized how trash I am at this (handle: shiningsunnyday). A lot to improve on here (after a few rounds now I always get As and B's for Div 2 quickly, and occasionally C's, so my goal is to get A-D every contest). I know C++ is the preferred language, but for some reason Java-C++ syntax just throws me off every time, so I'm going to stick with Python (I don't think it matters until a certain point when it gets too slow, but I think CF is decent enough to make sure none of the problems get TLE in each of the languages offered, though I may be wrong).

I also built my first neural network in Python! It classifies images as either cat or non-cat, up to 70% accuracy (actually it's just logistic regression without a single hidden layer). I've been doing this Deep Learning series on Coursera by deeplearning.ai and Andrew Ng, and I'm almost done with the first course (it's really awesome). Once I introduce hidden layers to built deeper neural nets the accuracy should go up to 80-90% (and I can probably build way cooler stuff than image classifiers), but we'll see.

I also did a lot of reading on the MBTI Personality Test. When I took it last year I got INTP (though my four cognitive functions are all pretty borderline), then got INTJ, and INFP each once this time around. By my understanding INTP are engineers who're good with details, whereas INTJ can understand complex theories and turn them into practical uses (this actually sounds more like me). I would think most AoPSers are INT_'s, though I may be wrong.

Finally, I spent a lot of time just laying around, rapping Kpop, and getting fat. Every person goes through this some time of the year, you know.

So I'm currently in the Bay Area, staying with my mom. ANYWAYS, 2018 is here and it'll be a fun year, and as opposed to listing a million things which I won't complete, here's my goals. Finally, to document a lotta the cool academic-related stuff I do from now on (book reviews, python projects, etc.) go check out the website shiningsunnyday.com!

goals

Submit

The blog is locked right?
by First, Apr 14, 2018, 6:00 PM
Great, amazing, inspiring blog. Good luck in life, and just know I aspire to succeed as you will in the future.
by mgrimalo, Apr 7, 2018, 6:19 PM
Yesyesyes
by shiningsunnyday, Mar 29, 2018, 5:30 PM
:O a new background picture
by MathAwesome123, Mar 29, 2018, 3:39 PM
did you get into MIT?
by 15Pandabears, Mar 15, 2018, 10:42 PM
wait what new site?
by yegkatie, Feb 11, 2018, 1:49 AM
Yea, doing a bit of cleaning before migrating to new site
by shiningsunnyday, Jan 21, 2018, 2:43 PM
Were there posts made in December 2017 for this blog and then deleted?

I ask because I was purging my thunderbird inbox and I found emails indicating new blog posts of yours.

email do not lie
by jonlin1000, Jan 21, 2018, 12:12 AM
@below sorry not accepting contribs
by shiningsunnyday, Dec 11, 2017, 11:15 AM
contrib plez?
also wow this blog is very popular
by DavidUsa, Dec 10, 2017, 7:53 PM
@First: lol same

first shout of december
by coolmath34, Dec 6, 2017, 2:32 PM
XD this blog is hilarious
by Mitsuku, Nov 21, 2017, 7:40 PM
@wu2481632: stop encouraging SSD to procrastinate(blog entries are fun but procrastination isn't).
by First, Aug 7, 2017, 5:02 PM
3.5 weeks without a post
by Flash12, Aug 4, 2017, 8:10 AM
First august shout!!
by adik7, Aug 1, 2017, 6:52 AM
"Also, I figured that whenever"

Darn, whenever what?

Belated (oops sorry) well-wishes on the 2016 AMC10A!
by DeathLlama9, Feb 3, 2016, 3:23 AM
what is a 8char?
lol 8char
by whiteawesomesun, Jan 30, 2016, 9:43 AM
lol 8char
by chocopuff, Jan 30, 2016, 3:38 AM
yadayadayadayadayadayada amc 10 is near... hope you get into JMO...
by whiteawesomesun, Jan 30, 2016, 2:45 AM
so much hype and stress!
by chocopuff, Jan 29, 2016, 2:14 PM
Lol lets reconcile man
by Konigsberg, Jan 20, 2016, 11:36 PM
DUDUUDDUDUUDE STAPHHHHH. war? war? TF boys
by whiteawesomesun, Jan 19, 2016, 8:50 AM
dude dude stop stop war war
by Konigsberg, Jan 18, 2016, 3:44 AM
I'm excited!
by paperplates, Jan 15, 2016, 6:55 AM
nice! \8char
by adihaya, Jan 14, 2016, 6:47 AM
yay, looks like this is going to be a cool blog!
by jh235, Jan 7, 2016, 8:15 PM
Post something!
by whiteawesomesun, Jan 4, 2016, 3:54 PM
at least I claim second shot
by Kline, Dec 29, 2015, 1:45 AM
first shout >
by oceanair, Dec 28, 2015, 6:00 AM

416 shouts

SSD's Musings and Dreams

by EthanWYX2009, Mar 18, 2025, 8:24 AM

by AlperenINAN, Mar 18, 2025, 6:27 AM

by swynca, Mar 18, 2025, 6:24 AM

by sqing, Mar 18, 2025, 4:16 AM

by sqing, Mar 17, 2025, 8:32 AM

by sqing, Mar 17, 2025, 8:15 AM

by Noname23, Mar 16, 2025, 4:57 PM

by sarjinius, Mar 9, 2025, 3:42 PM

by anantmudgal09, Mar 7, 2021, 10:33 AM

by GorgonMathDota, Nov 8, 2020, 12:52 AM

by shiningsunnyday, Jan 1, 2018, 12:06 PM