Mobius function

by luutrongphuc, Apr 28, 2025, 12:14 PM

Consider a sequence $(a_n)$ that satisfies:
$\sum_{i=1}^{n} a_{\left\lfloor \frac{n}{i} \right\rfloor} = n^k$
Let $c$ be a positive integer. Prove that for all integers $n > 1$ , we have:
$\frac{c^{a_n} - c^{a_{n-1}}}{n} \in \mathbb{Z}$

number theory

function

f(f(x)+y) = x+f(f(y))

by NicoN9, Apr 28, 2025, 10:03 AM

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(f(x)+y) = x+f(f(y))$ for all $x, y\in \mathbb{R}$ .

function

algebra

Function from the plane to the real numbers

by AndreiVila, Apr 28, 2025, 6:50 AM

Let $f:\pi\rightarrow\mathbb{R}$ be a function from the Euclidean plane to the real numbers such that $f(A)+f(B)+f(C)=f(O)+f(G)+f(H)$ for any acute triangle $ABC$ with circumcenter $O$ , centroid $G$ and orthocenter $H$ . Prove that $f$ is constant.

Inversion exercise

by Assassino9931, Apr 27, 2025, 10:29 PM

Let $ABC$ be an acute scalene triangle $ABC$ , $D$ be the orthogonal projection of $A$ on $BC$ , $M$ and $N$ are the midpoints of $AB$ and $AC$ respectively. Let $P$ and $Q$ are points on the minor arcs $\widehat{AB}$ and $\widehat{AC}$ of the circumcircle of triangle $ABC$ respectively such that $PQ \parallel BC$ . Show that the circumcircles of triangles $DPQ$ and $DMN$ are tangent if and only if $M$ lies on $PQ$ .

Inversion

BMO Shortlist

geometry

Vasc = 1?

by Li4, Apr 26, 2025, 1:33 PM

Find all integer tuples $(a, b, c)$ such that
$(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1.$
Proposed by Li4, Untro368, usjl and YaWNeeT.

number theory

inequalities

hard problem

by Cobedangiu, Apr 21, 2025, 1:51 PM

Let $a,b,c>0$ and $a+b+c=3$ . Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$

inequalities

Killer NT that nobody solved (also my hardest NT ever created)

by mshtand1, Apr 19, 2025, 9:31 PM

A positive integer number $a$ is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence $\{b_k\}_{k=1}^{\infty}$ , where
$b_k = a^{k^k} \cdot 2^{2^k - k} + 1.$
Proposed by Arsenii Nikolaev and Mykhailo Shtandenko

number theory

a_1 is anything but 2

by EeEeRUT, Jul 18, 2024, 4:05 PM

The sequence $(a_n)_{n\in\mathbb{N}}$ is defined by $a_1=3$ and $a_n=a_1a_2\cdots a_{n-1}-1$ Show that there exist infinitely many prime number that divide at least one number in this sequences

This post has been edited 2 times. Last edited by EeEeRUT, Jul 18, 2024, 4:08 PM
Reason: Idk

number theory

prime numbers

2021 EGMO P1: {m, 2m+1, 3m} is fantabulous

by anser, Apr 13, 2021, 12:00 PM

The number 2021 is fantabulous. For any positive integer $m$ , if any element of the set $\{m, 2m+1, 3m\}$ is fantabulous, then all the elements are fantabulous. Does it follow that the number $2021^{2021}$ is fantabulous?

Domain swept by Parabola

by Kunihiko_Chikaya, Jan 3, 2012, 5:53 AM

In the $x$ - $y$ plane, given a parabola $C_t$ passing through 3 points $P(t-1,\ t),\ Q(t,\ t)$ and $R(t+1,\ t+2)$ .
Let $t$ vary in the range of $-1\leq t\leq 1$ , draw the domain swept out by $C_t$ .

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 luutrongphuc, Apr 28, 2025, 12:14 PM

by NicoN9, Apr 28, 2025, 10:03 AM

by AndreiVila, Apr 28, 2025, 6:50 AM

by Assassino9931, Apr 27, 2025, 10:29 PM

by Li4, Apr 26, 2025, 1:33 PM

by Cobedangiu, Apr 21, 2025, 1:51 PM

by mshtand1, Apr 19, 2025, 9:31 PM

by EeEeRUT, Jul 18, 2024, 4:05 PM

by anser, Apr 13, 2021, 12:00 PM

by Kunihiko_Chikaya, Jan 3, 2012, 5:53 AM