BABBAGE'S THEOREM EXTENSION

by Mathgloggers, Apr 29, 2025, 12:18 PM

A few days ago I came across. this interesting result is someone interested in proving this.

$\boxed{\sum_{k=1}^{p-1} \frac{1}{k} \equiv \sum_{k=p+1}^{2p-1} \frac{1}{k} \equiv \sum_{k=2p+1}^{3p-1}\frac{1}{k} \equiv.....\sum_{k=p(p-1)+1}^{p^2-1}\frac{1}{k} \equiv 0(mod p^2)}$

This post has been edited 1 time. Last edited by Mathgloggers, Yesterday at 12:22 PM
Reason: D

number theory

Inequality, tougher than it looks

by tom-nowy, Apr 29, 2025, 3:51 AM

Prove that for $a,b \in \mathbb{R}$
$2(a^2+1)(b^2+1) \geq 3(a+b).$ Is there an elegant way to prove this?

Inequality

Dot product

by SomeonecoolLovesMaths, Apr 28, 2025, 6:06 PM

How to prove that dot product is distributive?

BrUMO 2025 Team Round Problem 3

by lpieleanu, Apr 27, 2025, 11:05 PM

Bruno and Brutus are running on a circular track with a $20$ foot radius. Bruno completes $5$ laps every hour, while Brutus completes $7$ laps every hour. If they start at the same point but run in opposite directions, how far along the track’s circumference (in feet) from the starting point are they when they meet for the sixth time? Note: Do not count the moment they start running as a meeting point.

BrUMO

Sequence

by lgx57, Apr 27, 2025, 12:56 PM

$a_1=1,a_{n+1}=a_n+\frac{1}{a_n}$ . Find the general term of $\{a_n\}$ .

algebra

Sequences

Inequalities

by sqing, Apr 26, 2025, 12:58 PM

Let $x\in(-1,1).$ Prove that
$\dfrac{1}{\sqrt{1-x^2}} + \dfrac{1}{2+ x^2} \geq \dfrac{3}{2}$ $\dfrac{2}{\sqrt{1-x^2}} + \dfrac{1}{1+x^2} \geq 3$

inequalities

algebra

Geometry Angle Chasing

by Sid-darth-vater, Apr 21, 2025, 11:50 PM

Is there a way to do this without drawing obscure auxiliary lines? (the auxiliary lines might not be obscure I might just be calling them obscure)

For example I tried rotating triangle MBC 80 degrees around point C (so the BC line segment would now lie on segment AC) but I couldn't get any results. Any help would be appreciated!

Attachments:

geometry

Inequlities

by sqing, Jul 19, 2024, 12:48 PM

Let $a,b,c\geq 0$ and $a^2+ab+bc+ca=3 .$ Prove that $\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2} \geq \frac{3}{2}$ $\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2}-bc \geq -\frac{3}{2}$

This post has been edited 1 time. Last edited by sqing, Jul 19, 2024, 12:59 PM

inequalities

algebra

Inequalities

by sqing, Dec 22, 2023, 4:06 AM

Let $a,b,c\geq 0,ab+bc+ca>0$ and $a+b+c=3$ . Prove that
$\frac{8}{3}\leq\frac{(a+b)(b+c)(c+a)}{ab+bc+ca}\leq 3$ $3\leq\frac{(a+b)(2b+c)(c+a)}{ab+bc+ca}\leq 6$ $\frac{3}{2}\leq\frac{(a+b)(2b+c)(c+ a)}{ab+bc+2ca}\leq 6$ $1\leq\frac{(a+b)(2b+c)(c+ a)}{ab+bc+ 3ca}\leq 6$

inequalities

algebra

N.S. condition of passing a fixed point for a function

by Kunihiko_Chikaya, Sep 6, 2009, 11:27 AM

Let $f(t)$ be a function defined in any real numbers $t$ with $f(0)\neq 0.$ Prove that on the $x-y$ plane, the line $l_t : tx+f(t) y=1$ passes through the fixed point which isn't on the $y$ axis in regardless of the value of $t$ if only if $f(t)$ is a linear function in $t$ .

function

Submit

I still exist as well.
by G.G.Otto, Aug 11, 2023, 2:44 AM
hello I'm still here lol
by player01, Aug 6, 2022, 6:24 PM
[REVIVAL] I will start posting more calculator relating posts very soon. Even though school has been busy, I have been programming my calculators a decent amount, so I have a lot to share...
by sonone, Feb 18, 2022, 10:29 PM
wow its been like 2.5 years since geo class
by pieMax2713, Feb 4, 2022, 8:38 PM
@violin21, I've been very busy with school lately and haven't been able to add another lesson. I will when i get a free moment
by sonone, Aug 19, 2021, 12:45 AM
ORZ CODER
by samrocksnature, Aug 9, 2021, 9:57 PM
Could you make more Asymptote lessons on your "How to do Asymptote" blog?
by violin21, Aug 9, 2021, 7:26 PM
You can take it, just C&P the CSS into your CSS area
by sonone, Apr 17, 2021, 10:08 PM
how can we take the CSS if we have permission to not take it?
by GoogleNebula, Apr 17, 2021, 5:22 PM
That is awesome!
by sonone, Apr 15, 2021, 10:09 PM
I modified your dodecahedron and got:
[asy]
import three;
import solids;
size(300);
currentprojection=orthographic(0,1.3,1.2);
light(0,5,10);

real phi=(sqrt(6)+1)/3;
real g=(phi-1)/2;
real s=1/2;
real a=sqrt(1-phi*phi/4-g*g)+phi/2;

triple[] d;
d[0]=(phi
by Andrew2019, Mar 26, 2021, 12:15 AM
Not too many, just changing the color here and there. I really like your CSS!
by sonone, Feb 2, 2021, 10:35 AM
Nice!

I see you're making changes to the CSS.
by G.G.Otto, Feb 1, 2021, 9:26 PM
I'm learning Java now!
by sonone, Feb 1, 2021, 5:56 PM
And I took part of it from CaptainFlint and then added a ton of modifications.
by G.G.Otto, Dec 1, 2020, 8:56 AM
fixed $\text{[math]}$
by sonone, Apr 25, 2020, 8:43 PM
Change in progress...
by sonone, Apr 25, 2020, 8:42 PM
When it says "At least 11 users clicked that button" it's incorrect since one person could have just posted 11 times. It should say "At most 11 users click that button".
by G.G.Otto, Apr 25, 2020, 8:26 PM
see this PM
by sonone, Apr 25, 2020, 7:03 PM
I can't get the cursor to work.
by G.G.Otto, Apr 25, 2020, 6:38 PM
If the cursor is confusing, where you are actually pointing in a front of the ship
by sonone, Apr 25, 2020, 11:04 AM
It's ok $\text{[math]}$
by sonone, Apr 23, 2020, 5:30 PM
Sorry I haven't been posting. I've got a Research Paper (for Roman History) due in a week.
by G.G.Otto, Apr 22, 2020, 11:15 PM
You should download it, as it's really useful when learning python. You can find out more here
by G.G.Otto, Apr 19, 2020, 10:04 PM
No. I don't have IDLE.
by sonone, Apr 18, 2020, 7:49 PM
Do you have IDLE installed on your computer? If so, I can give you a couple really cool codes that made that don't work in the AoPS widgets.
by G.G.Otto, Apr 14, 2020, 6:47 AM
I don't know how to change it....
by G.G.Otto, Apr 5, 2020, 5:58 PM
Do you know how the change the into thing above my avatar a different color? It's too hard to see.
by sonone, Apr 5, 2020, 2:08 PM
WELCOME TO MY BLOG!
by sonone, Mar 31, 2020, 4:56 PM

98 shouts

sonone's place

by Mathgloggers, Apr 29, 2025, 12:18 PM

by tom-nowy, Apr 29, 2025, 3:51 AM

by SomeonecoolLovesMaths, Apr 28, 2025, 6:06 PM

by lpieleanu, Apr 27, 2025, 11:05 PM

by lgx57, Apr 27, 2025, 12:56 PM

by sqing, Apr 26, 2025, 12:58 PM

by Sid-darth-vater, Apr 21, 2025, 11:50 PM

by sqing, Jul 19, 2024, 12:48 PM

by sqing, Dec 22, 2023, 4:06 AM

by Kunihiko_Chikaya, Sep 6, 2009, 11:27 AM