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

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.

Nice FE as the First Day Finale

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

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

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$$ \frac{3}{2}\le \frac{1}{a^2+1}+\frac{1}{b^2+1}\le 1+\frac{1 }{\sqrt2} $$

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

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

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.

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

Pieces // Class of 2022 // Final Update On This Blog [Part 1/3]

by shiningsunnyday, Mar 29, 2018, 2:09 PM

Rejected from Harvard, Princeton, Yale, MIT, UCLA, UCSD.
Your spots are the most coveted of any US college.
Yet you took a chance in me when nobody else did.
This means the world to me.
Destiny calls from the West.

Committed Stanford Class of 2022!
Pieces from memory

Harvard
New kid
Betrayal
Isolation
Garbage
Class clown
A bit too far
Awakening
Hustle begins
Lonelier the higher you get
Hello, anyone there?
Overnight sensation
Back when I was nothing.
Where were you guys again?
When I had nothing and stood bluffing.
Where were you back then?

Crazy they say my aspirations then,
Change they ask of me to fit in there,
Inspirational they say my aspirations are now,
Now people change to be more like me here.

Cause one's success comes at others' costs,
With scarcity there is competition, that's the natural law,
Which filters our behaviors through selfish thoughts,
Scarcity makes us products o' the system of us all.
Attachments:
This post has been edited 13 times. Last edited by shiningsunnyday, Apr 1, 2018, 6:13 AM

The ones who are crazy enough to think they can change the world are the ones who do.

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shiningsunnyday
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+ January 2018
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  • 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 :o

    by Flash12, Aug 4, 2017, 8:10 AM

  • First august shout!!

    by adik7, Aug 1, 2017, 6:52 AM

416 shouts
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