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

Interesting inequality

by sqing, Mar 18, 2025, 7:50 AM

Let $a,b\geq 2 .$ Prove that
$(a^2-1)(b^2-1) (1-ab)+\frac{27}{8}a^2b^2\leq 27$ $(a^2-1)(b^2-1)(1-a^2b^2 )+\frac{81}{4}a^2b^2 \leq 189$ $(a^2-1)(b^2-1)(1-a^2b^2 )+ 162 ab \leq 513$ $(a^2-1)(b^2-1) (1-a^2b^2 )+21 a^2b^2\leq \frac{3219}{16}$ $(a^2-1)(b^2-1) (1-ab)+\frac{27}{8}a^2b^2\leq\frac{415+61\sqrt{61}}{18}$

This post has been edited 1 time. Last edited by sqing, 2 hours ago

inequalities proposed

inequalities

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

Minimal Grouping in a Complete Graph

by swynca, Mar 18, 2025, 6:10 AM

In a complete graph with $2025$ vertices, each edge has one of the colors $r_1$ , $r_2$ , or $r_3$ . For each $i = 1,2,3$ , if the $2025$ vertices can be divided into $a_i$ groups such that any two vertices connected by an edge of color $r_i$ are in different groups, find the minimum possible value of $a_1 + a_2 + a_3$ .

graph theory

combinatorics

edge coloring

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

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

D1010 : How it is possible ?

by Dattier, Mar 10, 2025, 10:49 AM

Is it true that $\forall n \in \mathbb N^*, (24^n \times B \mod A) \mod 2 = 0$ ?

A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975

B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902

This post has been edited 6 times. Last edited by Dattier, Sunday at 10:10 AM

number theory

Computer solutions

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.

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
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@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
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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 sqing, Mar 18, 2025, 7:50 AM

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

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

by swynca, Mar 18, 2025, 6:10 AM

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

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

by Dattier, Mar 10, 2025, 10:49 AM

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

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