Involved conditional geo

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

Let $ABC$ be an acute-angled triangle with $AB < AC$ , orthocenter $H$ , circumcircle $\Gamma$ and circumcentre $O$ . Let $M$ be the midpoint of $BC$ and let $D$ be a point such that $ADOH$ is a parallellogram. Suppose that there exists a point $X$ on $\Gamma$ and on the opposite side of $DH$ to $A$ such that $\angle DXH + \angle DHA = 90^{\circ}$ . Let $Y$ be the midpoint of $OX$ . Prove that if $MY = OA$ , then $OA = 2OH$ .

geometry

orthocenter

conditional geometry

BMO Shortlist

Projections on collections of lines

by Assassino9931, Apr 27, 2025, 10:17 PM

Let $\mathcal{D}$ be the set of all lines in the plane and $A$ be a set of $17$ points in the plane. For a line $d\in \mathcal{D}$ let $n_d(A)$ be the number of distinct points among the orthogonal projections of the points from $A$ on $d$ . Find the maximum possible number of distinct values of $n_d(A)$ (this quantity is computed for any line $d$ ) as $A$ varies.

This post has been edited 1 time. Last edited by Assassino9931, 5 hours ago

combinatorics

combinatorial geometry

BMO Shortlist

Good Permutations in Modulo n

by swynca, Apr 27, 2025, 2:03 PM

An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$ , such that:
$(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$ ;
$(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$ .
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.

This post has been edited 2 times. Last edited by swynca, Yesterday at 4:15 PM

BMO

number theory

abstract algebra

weird Condition

by B1t, Apr 27, 2025, 1:37 PM

In triangle $ABC$ , where $AC < AB$ , the internal angle bisectors of angles $\angle A$ , $\angle B$ , and $\angle C$ meet the sides $BC$ , $AC$ , and $AB$ at points $D$ , $E$ , and $F$ , respectively. Let $I$ be the incenter of triangle $AEF$ , and let $G$ be the foot of the perpendicular from $I$ to line $BC$ . Prove that if the quadrilateral $DGEF$ is cyclic, then the center of its circumcircle lies on segment $AD$ .

This post has been edited 4 times. Last edited by B1t, Yesterday at 5:28 PM

geometry

all functions satisfying f(x+yf(x))+y = xy + f(x+y)

by falantrng, Apr 27, 2025, 11:52 AM

Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ ,
$f(x+yf(x))+y = xy + f(x+y).$
Proposed by Giannis Galamatis, Greece

This post has been edited 1 time. Last edited by falantrng, Yesterday at 12:02 PM
Reason: added author

functional equation

function

algebra

Arbitrary point on BC and its relation with orthocenter

by falantrng, Apr 27, 2025, 11:47 AM

In an acute-angled triangle $ABC$ , $H$ be the orthocenter of it and $D$ be any point on the side $BC$ . The points $E, F$ are on the segments $AB, AC$ , respectively, such that the points $A, B, D, F$ and $A, C, D, E$ are cyclic. The segments $BF$ and $CE$ intersect at $P.$ $L$ is a point on $HA$ such that $LC$ is tangent to the circumcircle of triangle $PBC$ at $C.$ $BH$ and $CP$ intersect at $X$ . Prove that the points $D, X,$ and $L$ lie on the same line.

Proposed by Theoklitos Parayiou, Cyprus

This post has been edited 1 time. Last edited by falantrng, Yesterday at 4:38 PM

geometry

circumcircle

Interesting inequalities

by sqing, Apr 27, 2025, 3:12 AM

Let $a,b\geq 0$ and $a+b+ab=3.$ Prove that
$ab^2( b +1) \leq 4$ $ab( b +1) \leq \frac{9}{4}$ $a^2b ( a+b^2 ) \leq \frac{76}{27}$ $a^2b( b +1 ) \leq \frac{3(69-11\sqrt{33})}{8}$ $a^2b^2( b +1 ) \leq \frac{2(73\sqrt{73}-595)}{27}$

This post has been edited 1 time. Last edited by sqing, Yesterday at 4:53 AM

inequalities proposed

inequalities

Beautiful Number Theory

by tastymath75025, Jul 9, 2023, 4:42 AM

Prove that $5^n-3^n$ is not divisible by $2^n+65$ for any positive integer $n$ .

This post has been edited 1 time. Last edited by tastymath75025, Jul 9, 2023, 4:20 PM
Reason: fix wording

number theory

Easy way to delete your post in another user's blog

by sonone, May 17, 2021, 6:57 PM

Here is a really easy way to delete your post in another user's blog.

Step 1: Inspect Element
Right click on the report button, and click "Inspect Element". h

Step 2: Locate line
There should be a highlighted line. Right click and do to "Edit", then click "HTML". h

Step 3: Edit HTML
Change "blog-report-post" to "blog-delete-post". h

Step 4: Click the "Report" button
Click out of the editing area and click "Repot". h

Step 5: Delete post
Click "OK" and POOF!, your post is gone. h

You can also edit a post in a similar fashion.

This post has been edited 1 time. Last edited by sonone, May 17, 2021, 7:00 PM

shortcut

Fun

Infimum of decreasing sequence b_n/n^2

by a1267ab, Dec 16, 2019, 5:08 PM

Choose positive integers $b_1, b_2, \dotsc$ satisfying
$1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb$ and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$ . What are the possible values of $r$ across all possible choices of the sequence $(b_n)$ ?

Carl Schildkraut and Milan Haiman

This post has been edited 3 times. Last edited by a1267ab, Dec 16, 2019, 6:11 PM

algebra

The three lines AA', BB' and CC' meet on the line IO

by WakeUp, Mar 3, 2012, 8:01 PM

Let $ABC$ be a triangle and let $I$ and $O$ denote its incentre and circumcentre respectively. Let $\omega_A$ be the circle through $B$ and $C$ which is tangent to the incircle of the triangle $ABC$ ; the circles $\omega_B$ and $\omega_C$ are defined similarly. The circles $\omega_B$ and $\omega_C$ meet at a point $A'$ distinct from $A$ ; the points $B'$ and $C'$ are defined similarly. Prove that the lines $AA',BB'$ and $CC'$ are concurrent at a point on the line $IO$ .

(Russia) Fedor Ivlev

geometry

incenter

circumcircle

geometric transformation

reflection

homothety

trigonometry

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