euler-totient function

by Laan, Mar 21, 2025, 7:13 AM

Proof that there are infinitely many positive integers $n$ such that
$\varphi(n)<\varphi(n+1)<\varphi(n+2)$

number theory

Function Equation

by Dynic, Mar 21, 2025, 6:05 AM

Find all $f:\mathbb{R} \to \mathbb{R}$ such that
$f(x-f(y))=f(x+f(y)+y^5)+f(2f(y)+y^5)+2025,\forall x,y\in \mathbb{R}$

function

Cubic function from Olymon

by Adywastaken, Mar 21, 2025, 5:25 AM

Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
$x^2y^2(f(x+y)-f(x)-f(y))=3(x+y)f(x)f(y)$ $\forall$ $x,y \in \mathbb{R}$

This post has been edited 2 times. Last edited by Adywastaken, 3 hours ago

functional equation

algebra

function

Checkerboard

by Ecrin_eren, Mar 21, 2025, 5:20 AM

On an 8×8 checkerboard, what is the minimum number of squares that must be marked (including the marked ones) so that every square has exactly one marked neighbor? (We define neighbors as squares that share a common edge, and a square is not considered a neighbor of itself.)

This post has been edited 1 time. Last edited by Ecrin_eren, 3 hours ago
Reason: Checkerboard

combinatorics unsolved

combinatorics

Inspired by my own results

by sqing, Mar 21, 2025, 4:17 AM

Let $a,b,c\geq 2.$ Prove that
$(a+1)(b+2)(c +1)-3 abc\leq 12$ $(a+1)(b+2)(c +1)-\frac{7}{2}abc\leq 8$ $(a+1)(b+3)(c +1)-\frac{15}{4}abc\leq 15$ $(a+1)(b+3)(c +1)-4abc\leq 13$

inequalities proposed

algebra

Number Theory

by MuradSafarli, Mar 20, 2025, 7:55 PM

find all natural numbers $(a, b)$ such that the following equation holds:

$7^a + 1 = 2b^2$

number theory

Mathhhhh

by mathbetter, Mar 20, 2025, 11:21 AM

Three turtles are crawling along a straight road heading in the same
direction. "Two other turtles are behind me," says the first turtle. "One turtle is
behind me and one other is ahead," says the second. "Two turtles are ahead of me
and one other is behind," says the third turtle. How can this be possible?

This post has been edited 1 time. Last edited by mathbetter, Yesterday at 11:21 AM

combinatorics

Finally hard NT on UKR MO from NT master

by mshtand1, Mar 13, 2025, 11:42 PM

A pair of positive integer numbers $(a, b)$ is given. It turns out that for every positive integer number $n$ , for which the numbers $(n - a)(n + b)$ and $n^2 - ab$ are positive, they have the same number of divisors. Is it necessarily true that $a = b$ ?

Proposed by Oleksii Masalitin

number theory

Colouring numbers

by kitun, Nov 15, 2021, 5:50 AM

What is the least number required to colour the integers $1, 2,.....,2^{n}-1$ such that for any set of consecutive integers taken from the given set of integers, there will always be a colour colouring exactly one of them? That is, for all integers $i, j$ such that $1<=i<=j<=2^{n}-1$ , there will be a colour coloring exactly one integer from the set $i, i+1,.... , j-1, j$ .

This post has been edited 1 time. Last edited by kitun, Nov 15, 2021, 10:12 AM

Discrete Mathematics

number theory

Sequence bounding itself

by juckter, Jan 13, 2021, 4:46 AM

We say a sequence of integers $a_1, a_2, \dots, a_n$ is self-bounded if for each $i$ , $1 \le i \le n$ there exist at least $a_i$ terms of the sequence that are less than or equal to $i$ . Find the maximum possible value of $a_1 + a_2 + \dots + a_n$ for a self-bounded sequence $a_1, a_2, \dots, a_n$ .

This post has been edited 1 time. Last edited by juckter, Jan 13, 2021, 4:57 AM

combinatorics

Integer sequence

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 Laan, Mar 21, 2025, 7:13 AM

by Dynic, Mar 21, 2025, 6:05 AM

by Adywastaken, Mar 21, 2025, 5:25 AM

by Ecrin_eren, Mar 21, 2025, 5:20 AM

by sqing, Mar 21, 2025, 4:17 AM

by MuradSafarli, Mar 20, 2025, 7:55 PM

by mathbetter, Mar 20, 2025, 11:21 AM

by mshtand1, Mar 13, 2025, 11:42 PM

by kitun, Nov 15, 2021, 5:50 AM

by juckter, Jan 13, 2021, 4:46 AM