Do not try to bash on beautiful geometry

by ItzsleepyXD, Apr 30, 2025, 9:30 AM

Let $ABC$ be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$

geometry

Trivial fun Equilateral

by ItzsleepyXD, Apr 30, 2025, 9:05 AM

Let $ABC$ be a scalene triangle with point $P$ and $Q$ on the plane such that $\triangle BPC , \triangle CQB$ is an equilateral . Let $AB$ intersect $CP$ and $CQ$ at $X$ and $Z$ respectively and $AC$ intersect $BP$ and $BQ$ at $Y$ and $W$ respectively .
Prove that $XY\parallel ZW$

geometry

Equilateral

problem interesting

by Cobedangiu, Apr 30, 2025, 5:06 AM

Let $a=3k^2+3k+1 (a,k \in N)$
$i)$ Prove that: $a^2$ is the sum of $3$ square numbers
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers

This post has been edited 1 time. Last edited by Cobedangiu, Today at 5:06 AM

algebra

Question on Balkan SL

by Fmimch, Apr 30, 2025, 12:13 AM

Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!

Geometry with orthocenter config

by thdnder, Apr 29, 2025, 5:26 PM

Let $ABC$ be a triangle, and let $AD, BE, CF$ be its altitudes. Let $H$ be its orthocenter, and let $O_B$ and $O_C$ be the circumcenters of triangles $AHC$ and $AHB$ . Let $G$ be the second intersection of the circumcircles of triangles $FDO_B$ and $EDO_C$ . Prove that the lines $DG$ , $EF$ , and $A$ -median of $\triangle ABC$ are concurrent.

Hard inequality

by JK1603JK, Apr 29, 2025, 4:24 AM

Let $a,b,c>0$ and $a^2+b^2+c^2=2(a+b+c).$ Find the minimum $P=(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$

inequalities

Something about (BIC)

by flower417477, Apr 28, 2025, 3:58 PM

Given $\triangle ABC$ with incenter $I$ , $D$ is a point on $BC$ ,the bisector of $\angle ADB$ meet $(BIC)$ at $E,F$ .Prove that $\angle EAD=\angle IAF$

geometry

incenter

Olympiad

easy functional

by B1t, Apr 26, 2025, 6:45 AM

Denote the set of real numbers by $\mathbb{R}$ . Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all $x, y, z \in \mathbb{R}$ ,
$f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).$

This post has been edited 2 times. Last edited by B1t, Apr 26, 2025, 7:01 AM

algebra

Choose P on (AOB) and Q on (AOC)

by MarkBcc168, Jun 22, 2024, 3:54 PM

Let $ABC$ be a triangle with circumcenter $O$ and circumcircle $\omega$ . Let $D$ be the foot of the altitude from $A$ to $\overline{BC}$ . Let $P$ and $Q$ be points on the circumcircles of triangles $AOB$ and $AOC$ , respectively, such that $A$ , $P$ , and $Q$ are collinear. Prove that if the circumcircle of triangle $OPQ$ is tangent to $\omega$ at $T$ , then $\angle BTD=\angle CAP$ .

Tiger Zhang

geometry

7^a - 3^b divides a^4 + b^2 (from IMO Shortlist 2007)

by Dida Drogbier, Apr 21, 2008, 2:53 PM

Find all pairs of natural numbers $(a, b)$ such that $7^a - 3^b$ divides $a^4 + b^2$ .

Author: Stephan Wagner, Austria

Divisibility

IMO Shortlist

number theory

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 ItzsleepyXD, Apr 30, 2025, 9:30 AM

by ItzsleepyXD, Apr 30, 2025, 9:05 AM

by Cobedangiu, Apr 30, 2025, 5:06 AM

by Fmimch, Apr 30, 2025, 12:13 AM

by thdnder, Apr 29, 2025, 5:26 PM

by JK1603JK, Apr 29, 2025, 4:24 AM

by flower417477, Apr 28, 2025, 3:58 PM

by B1t, Apr 26, 2025, 6:45 AM

by MarkBcc168, Jun 22, 2024, 3:54 PM

by Dida Drogbier, Apr 21, 2008, 2:53 PM