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

Angles in Geometry done correctly

by kamatadu, Nov 2, 2024, 7:35 PM

POST BY AVIGYAN

This is going to be a very short blog-post discussing about the notations of angles.

Backstory: This blog-post is written because the notation $\measuredangle (\ell_1, \ell_2)$ confused me so much that I had to ask what it really means in an MSE post. I was still not satisfied with the explanation I got there. As I read a few more handouts, the notations became a bit more clear to me. I hope the idea I am sharing in this blog-post is indeed the correct one.

We start with vanilla angles first. Let's say we have three points $A$ , $B$ and $C$ . Then suppose we want to find the value of the angle $\angle ABC$ . We can extend the lines $AB$ and $BC$ to note that there are four angles that are formed. Which one do we take?

$[asy] /* Converted from GeoGebra by User:Azjps using Evan's magic cleaner https://github.com/vEnhance/dotfiles/blob/main/py-scripts/export-ggb-clean-asy.py */ /* A few re-additions are done using bubu-asy.py. This adds the dps, xmin, linewidth, fontsize and directions. https://github.com/Bubu-Droid/dotfiles/blob/master/bubu-scripts/bubu-asy.py */ pair A = (31.64835,16.28958); pair B = (40.11708,2.37181); pair C = (44.07794,21.53331); import graph; size(8cm); pen dps = linewidth(0.5) + fontsize(13); defaultpen(dps); real xmin = -5, xmax = 5, ymin = -5, ymax = 5; draw(arc(B,4.60985,78.32093,121.31981)--B--cycle, linewidth(0.6) + blue); draw(arc(B,4.60985,-101.67906,-58.68018)--B--cycle, linewidth(0.6) + blue); draw(arc(B,2.56591,-58.68018,78.32093)--B--cycle, linewidth(0.6) + red); draw(arc(B,2.56591,121.31981,258.32093)--B--cycle, linewidth(0.6) + red); draw(A--B, linewidth(0.6)); draw(B--C, linewidth(0.6)); draw((37.77614,-8.95301)--B, linewidth(0.6) + linetype("4 4")); draw(B--(45.63427,-6.69530), linewidth(0.6) + linetype("4 4")); draw(arc(B,4.60985,78.32093,121.31981), linewidth(0.6) + blue); draw(arc(B,4.27057,78.32093,121.31981), linewidth(0.6) + blue); draw(arc(B,4.60985,-101.67906,-58.68018), linewidth(0.6) + blue); draw(arc(B,4.27057,-101.67906,-58.68018), linewidth(0.6) + blue); dot("$A$", A, NW); dot("$B$", B, dir(180)); dot("$C$", C, N); [/asy]$

So here, we define the notation $\angle ABC$ as the magnitude of the angle with measure $\le 90^{\circ}$ angle of the four angles we found earlier. For this case, it is the blue angle.

But what does the notation $\angle (\ell_1,\ell_2)$ mean? Is it the one which is $\le 90^{\circ}$ or the other one? Does orientation matter?

This is where the doubt first comes up. In vanilla angles however, this question makes no sense as we always define the vanilla angle between two lines as the angle which is $\le 90^{\circ}$ . Vanilla angles do not care about orientation too.

Thus we can now safely conclude that $\angle (\ell_1,\ell_2) = \angle (\ell_2,\ell_1)$ . So far so good.

Before we begin, we should keep in mind that "Anti-clockwise is positive and clockwise is negative".

Let us now move onto directed angles. This is where things start behaving differently because suddenly, reflexive angles start being considered modulo $180^{\circ}$ and orientation also comes into play.

We begin with the same configuration. Let's say we have three points $A$ , $B$ and $C$ (where $A$ , $B$ and $C$ are in anti-clockwise order). Now we want to find the value of the directed angle $\measuredangle ABC$ .

$[asy] /* Converted from GeoGebra by User:Azjps using Evan's magic cleaner https://github.com/vEnhance/dotfiles/blob/main/py-scripts/export-ggb-clean-asy.py */ /* A few re-additions are done using bubu-asy.py. This adds the dps, xmin, linewidth, fontsize and directions. https://github.com/Bubu-Droid/dotfiles/blob/master/bubu-scripts/bubu-asy.py */ pair A = (31.64835,16.28958); pair B = (40.11708,2.37181); pair C = (44.07794,21.53331); import graph; size(8cm); pen dps = linewidth(0.5) + fontsize(13); defaultpen(dps); real xmin = -5, xmax = 5, ymin = -5, ymax = 5; draw(arc(B,4.60985,78.32093,121.31981)--B--cycle, linewidth(0.6) + blue); draw(arc(B,4.60985,-101.67906,-58.68018)--B--cycle, linewidth(0.6) + blue); draw(arc(B,2.56591,-58.68018,78.32093)--B--cycle, linewidth(0.6) + red); draw(arc(B,2.56591,121.31981,258.32093)--B--cycle, linewidth(0.6) + red); draw(A--B, linewidth(0.6)); draw(B--C, linewidth(0.6)); draw((37.77614,-8.95301)--B, linewidth(0.6) + linetype("4 4")); draw(B--(45.63427,-6.69530), linewidth(0.6) + linetype("4 4")); draw(arc(B,4.60985,78.32093,121.31981), linewidth(0.6) + blue,BeginArcArrow(6)); draw(arc(B,4.60985,-101.67906,-58.68018), linewidth(0.6) + blue); draw(arc(B,4.27057,-101.67906,-58.68018), linewidth(0.6) + blue); draw(arc(B,2.56591,121.31981,251.28453), linewidth(0.6) + red,EndArcArrow(6)); dot("$A$", A, NW); dot("$B$", B, dir(180)); dot("$C$", C, N); [/asy]$

For doing this, we take the line $BA$ and rotate it in an anti-clockwise direction w.r.t. $B$ until it gets super-imposed on the line $BC$ . (If you thought it was the other way around, you were wrong!)

The orientation does not depend on the ordering of the vertices but rather on the direction of rotation. (I had the opposite idea a few weeks back too!) This statement might feel a bit ambiguous right now, but just hold on.

So when we are finding the value of $\measuredangle ABC$ (as shown in the image above), the positive value of the directed angle is $> 90^{\circ}$ (the red angle) for the case in the image. Now if we take this value modulo $180^{\circ}$ to make the magnitude $\le 90^{\circ}$ , we find that the value is actually negative.

The anti-clockwise ordering of the vertices do not matter here, but rather the direction of rotation matters here. In order to get an angle which is $\le 90^{\circ}$ in magnitude, we make the rotation clockwise which results in the value of the angle being negative.

Now in the figure as shown below, $A$ , $B$ and $C$ are ordered in clockwise order. In this case, the value of $\measuredangle ABC$ is actually positive. (The ordering of the vertices does not matter. Only the direction of rotation does.)

$[asy] /* Converted from GeoGebra by User:Azjps using Evan's magic cleaner https://github.com/vEnhance/dotfiles/blob/main/py-scripts/export-ggb-clean-asy.py */ /* A few re-additions are done using bubu-asy.py. This adds the dps, xmin, linewidth, fontsize and directions. https://github.com/Bubu-Droid/dotfiles/blob/master/bubu-scripts/bubu-asy.py */ pair A = (31.64835,16.28958); pair B = (40.11708,2.37181); pair C = (44.07794,21.53331); import graph; size(8cm); pen dps = linewidth(0.5) + fontsize(13); defaultpen(dps); real xmin = -5, xmax = 5, ymin = -5, ymax = 5; draw(arc(B,4.60985,78.32093,121.31981)--B--cycle, linewidth(0.6) + blue); draw(arc(B,4.60985,-101.67906,-58.68018)--B--cycle, linewidth(0.6) + blue); draw(arc(B,2.56591,-58.68018,78.32093)--B--cycle, linewidth(0.6) + red); draw(arc(B,2.56591,121.31981,258.32093)--B--cycle, linewidth(0.6) + red); draw(A--B, linewidth(0.6)); draw(B--C, linewidth(0.6)); draw((37.77614,-8.95301)--B, linewidth(0.6) + linetype("4 4")); draw(B--(45.63427,-6.69530), linewidth(0.6) + linetype("4 4")); draw(arc(B,4.60985,78.32093,121.31981), linewidth(0.6) + blue,EndArcArrow(6)); draw(arc(B,4.60985,-101.67906,-58.68018), linewidth(0.6) + blue); draw(arc(B,4.27057,-101.67906,-58.68018), linewidth(0.6) + blue); draw(arc(B,2.56591,121.31981,251.28453), linewidth(0.6) + red); dot("$C$", A, NW); dot("$B$", B, dir(180)); dot("$A$", C, N); [/asy]$

To make this even more clear, we take an example from EGMO. This example should make the idea even more vivid.

Now let us get to the original question due to which this blog-post was made. How do we define $\measuredangle (\ell_1, \ell_2)$ ? If you've read through the blog-post till here, you should try this little exercise yourself first before moving on.

For doing this, we define $T=\ell_1\cap \ell_2$ . Now we rotate $\ell_1$ w.r.t. $T$ in an anti-clockwise direction till it gets super-imposed on the line $\ell_2$ . Now if the value of the angle is $\le 90^{\circ}$ , that's basically our desired directed angle and the sign also remains positive (since we rotated in an anti-clockwise direction). However, if the value of the angle turns out to be $\ge 90^{\circ}$ , then we take it modulo $180^{\circ}$ and make its magnitude $\le 90^{\circ}$ . In this case, the sign of the angle is negative. Thus this case is equivalent to taking the line $\ell_1$ and rotating it about $T$ in a clockwise direction till it gets super-imposed on $\ell_2$ .

$[asy] /* Converted from GeoGebra by User:Azjps using Evan's magic cleaner https://github.com/vEnhance/dotfiles/blob/main/py-scripts/export-ggb-clean-asy.py */ /* A few re-additions are done using bubu-asy.py. This adds the dps, xmin, linewidth, fontsize and directions. https://github.com/Bubu-Droid/dotfiles/blob/master/bubu-scripts/bubu-asy.py */ import graph; size(8cm); pen dps = linewidth(0.5) + fontsize(13); defaultpen(dps); real xmin = -5, xmax = 5, ymin = -5, ymax = 5; draw(arc((40.11708,2.37181),5.21970,78.32093,121.31981)--(40.11708,2.37181)--cycle, linewidth(0.6) + blue); draw(arc((62.92721,4.40749),5.21970,78.32093,121.31981)--(62.92721,4.40749)--cycle, linewidth(0.6) + red); draw((54.45847,18.32526)--(62.92721,4.40749), linewidth(0.6)); label("$\ell_1$", (54.45847,18.32526)--(62.92721,4.40749), SW); draw((62.92721,4.40749)--(66.88806,23.56899), linewidth(0.6)); label("$\ell_2$", (62.92721,4.40749)--(66.88806,23.56899), SE); draw((60.58626,-6.91732)--(62.92721,4.40749), linewidth(0.6)); draw((62.92721,4.40749)--(68.44439,-4.65961), linewidth(0.6)); draw((31.64835,16.28958)--(40.11708,2.37181), linewidth(0.6)); label("$\ell_2$", (31.64835,16.28958)--(40.11708,2.37181), SW); draw((40.11708,2.37181)--(44.07794,21.53331), linewidth(0.6)); label("$\ell_1$", (40.11708,2.37181)--(44.07794,21.53331), SE); draw((37.77614,-8.95301)--(40.11708,2.37181), linewidth(0.6)); draw((40.11708,2.37181)--(45.63427,-6.69530), linewidth(0.6)); draw(arc((40.11708,2.37181),5.21970,78.32093,119.21372), linewidth(0.6) + blue,EndArcArrow(6)); draw(arc((62.92721,4.40749),5.21970,78.32093,119.21372), linewidth(0.6) + red,BeginArcArrow(6)); [/asy]$

Thus we can conclude our findings for the directed angles as, $\measuredangle (\ell_1,\ell_2)=-\measuredangle (\ell_2,\ell_1)$ and
$\measuredangle (AB,BC)=\measuredangle ABC=-\measuredangle CBA=-\measuredangle (CB,BA) .$
However I should mention that $\measuredangle (AB,BC)$ is the same as $\measuredangle (AB,CB)$ since both the lines $BC$ and $CB$ are basically equivalent.

I hope I was able to convey my message through this blog-post (plix halp this nub to larn bemter englis :maybe:

).

Thank You!!

This post has been edited 8 times. Last edited by kamatadu, Nov 2, 2024, 7:51 PM

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

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

orz saar
by Levieee, Apr 16, 2025, 9:02 PM
i feel like i see a shoutout
by Nuterrow, Feb 27, 2025, 4:51 PM
Thanks for the tips
by math_holmes15, Feb 26, 2025, 6:13 AM
Good Luck with your new series
by SomeonecoolLovesMaths, Feb 24, 2025, 11:38 AM
hawsoworzz plij gibb teeps on how to attains skeels
by L13832, Nov 25, 2024, 6:53 AM
E bhai hothat cs change korli ki byapar
by HoRI_DA_GRe8, Nov 3, 2024, 8:03 PM
Ngl that pfp is orz
by factcheck315, Nov 2, 2024, 9:09 PM
finally shouting out orz ppl in orz blog
by Kappa_Beta_725, Oct 25, 2024, 6:13 PM
Omega lavil
Pro lavil
God lavil
Lavil lavil

by Sreemani76, Apr 19, 2024, 7:20 PM
Nice
by Om245, Jan 12, 2024, 2:18 AM
@below We lost finals, so GG...
by kamatadu, Nov 26, 2023, 2:43 PM
We won semis
by Project_Donkey_into_M4, Nov 15, 2023, 6:05 PM
yess sss
by polynomialian, Nov 3, 2023, 3:10 PM
We had a great camp in ISI back in February for MTRP (MATHEMATICS TALENT REWARD PROGRAM). So since the fest season of CMI and isi is near,this means the season is back again!!!.So should we post the mtrp 2023 experience?

Comment!
by HoRI_DA_GRe8, Nov 2, 2023, 4:27 PM
Saar good luck for RMO
haw prep going btw
Me dead non geo :skull:
by BVKRB-, Oct 20, 2023, 2:06 PM
ooo you also trying sharygin poblams nais saar, tell progress lol
by BVKRB-, Jul 5, 2022, 5:36 AM
@orsmax_viewer_sar, me nubmax is trying the sharygin poblams so me numbax not get time to update blog for few days :")
by kamatadu, Jul 3, 2022, 5:20 PM
@below no sar, I waste too much time sar :")
by kamatadu, Jul 1, 2022, 3:25 PM
hello sar bhul gaye kya
by anurag27826, Jun 28, 2022, 2:19 PM
AT LAST!!! Post on SET THEORY is finally complete
by kamatadu, Jun 21, 2022, 3:15 PM
IM BACK! Check the latest post for the description of why I went missing
by kamatadu, Jun 15, 2022, 6:09 PM
Guys im really sorry for not updating the blog for so long, actually ive been busy with fixing my mobile since the past few days, so I couldnt update my blog, I believe I will be done with this work by tomorrow and Ill again start updating the blog
by kamatadu, Jun 13, 2022, 11:24 AM
No bengali plx me no understand single word and me too lazy to use gugul translat
by BVKRB-, Jun 7, 2022, 8:55 AM
Amar dadu bhola dadu bhalo dadu noye.
Kamatadu orz
by Bratin_Dasgupta, May 29, 2022, 6:19 AM
hello sar
by anurag27826, May 26, 2022, 1:09 PM
ughh, my time management is literally the worst, cant even figure out time for updating blog
by kamatadu, May 25, 2022, 6:22 PM
ok saar enuf orsing
by BVKRB-, Mar 4, 2022, 5:39 PM
hello sar, plis domt depress by showing ur pure ors skills here pls
by kamatadu, Feb 27, 2022, 3:58 PM
"Hello World?"
by BVKRB-, Feb 27, 2022, 9:57 AM