Hagge-like circles, Jerabek hyperbola, Lemoine cubic

by kosmonauten3114, Jun 2, 2025, 4:05 PM

Let $\triangle{ABC}$ be a scalene oblique triangle with circumcenter $O$ and orthocenter $H$ , and $P$ ( $\neq \text{X(3), X(4)}$ , $\notin \odot(ABC)$ ) a point in the plane.
Let $\triangle{A_1B_1C_1}$ , $\triangle{A_2B_2C_2}$ be the circumcevian triangles of $O$ , $P$ , respectively.
Let $\triangle{P_AP_BP_C}$ be the pedal triangle of $P$ with respect to $\triangle{ABC}$ .
Let $A_1'$ be the reflection in $P_A$ of $A_1$ . Define $B_1'$ , $C_1'$ cyclically.
Let $A_2'$ be the reflection in $P_A$ of $A_2$ . Define $B_2'$ , $C_2'$ cyclically.
Let $O_1$ , $O_2$ be the circumcenters of $\triangle{A_1'B_1'C_1'}$ , $\triangle{A_2'B_2'C_2'}$ , respectively.

Prove that:
1) $P$ , $O_1$ , $O_2$ are collinear if and only if $P$ lies on the Jerabek hyperbola of $\triangle{ABC}$ .
2) $H$ , $O_1$ , $O_2$ are collinear if and only if $P$ lies on the Lemoine cubic (= $\text{K009}$ ) of $\triangle{ABC}$ .

Attachments:

geometry

reflection

Jerabek Hyperbola

Lemoine cubic

collinear

S(an) greater than S(n)

by ilovemath0402, Jun 2, 2025, 3:23 PM

Find all positive integer $n$ such that $S(an)\ge S(n) \quad \forall a \in \mathbb{Z}^{+}$ ( $S(n)$ is sum of digit of $n$ in base 10)
P/s: Original problem

sum of digits

number theory

arithmetric function

Parallel lines on a rhombus

by buratinogigle, Jun 2, 2025, 3:17 PM

Given the rhombus $ABCD$ with its incircle $\omega$ . Let $E$ and $F$ be the points of tangency of $\omega$ with $AB$ and $AC$ respectively. On the edges $CB$ and $CD$ , take points $G$ and $H$ such that $GH$ is tangent to $\omega$ at $P$ . Suppose $Q$ is the intersection point of the lines $EG$ and $FH$ . Prove that two lines $AP$ and $CQ$ are parallel or coincide.

Attachments:

geometry

rhombus

Line bisects a segment

by buratinogigle, Jun 2, 2025, 3:08 PM

Let $ABC$ be a triangle with $AB = AC$ . A circle $(O)$ is tangent to sides $AC$ and $AB$ , and $O$ is the midpoint of $BC$ . Points $E$ and $F$ lie on sides $AC$ and $AB$ , respectively, such that segment $EF$ is tangent to circle $(O)$ at point $P$ . Let $H$ and $K$ be the orthocenters of triangles $OBF$ and $OCE$ , respectively. Prove that line $OP$ bisects segment $HK$ .

Attachments:

geometry

FE inequality from Iran

by mojyla222, Apr 19, 2025, 9:20 AM

Find all functions $f:\mathbb{R}^+ \to \mathbb{R}$ such that for all $x,y,z>0$
$3(x^3+y^3+z^3)\geq f(x+y+z)\cdot f(xy+yz+xz) \geq (x+y+z)(xy+yz+xz).$

inequalities

function

Functional inequality

Iran 2nd Round

Tangency of circles with "135 degree" angles

by Shayan-TayefehIR, May 19, 2024, 3:50 PM

For a triangle $\triangle ABC$ with an obtuse angle $\angle A$ , let $E , F$ be feet of altitudes from $B , C$ on sides $AC , AB$ respectively. The tangents from $B , C$ to circumcircle of triangle $\triangle ABC$ intersect line $EF$ at points $K , L$ respectively and we know that $\angle CLB=135$ . Point $R$ lies on segment $BK$ in such a way that $KR=KL$ and let $S$ be a point on line $BK$ such that $K$ is between $B , S$ and $\angle BLS=135$ . Prove that the circle with diameter $RS$ is tangent to circumcircle of triangle $\triangle ABC$ .

Proposed by Mehran Talaei

This post has been edited 2 times. Last edited by Shayan-TayefehIR, May 27, 2024, 10:11 AM

geometry

Removing Numbers On A Blackboard

by Kezer, Aug 7, 2017, 9:43 PM

The numbers $1,2,3,\dots,2017$ are on the blackboard. Amelie and Boris take turns removing one of those until only two numbers remain on the board. Amelie starts. If the sum of the last two numbers is divisible by $8$ , then Amelie wins. Else Boris wins. Who can force a victory?

This post has been edited 1 time. Last edited by Kezer, Aug 7, 2017, 9:45 PM

blackboard

combinatorics

combinatorics unsolved

number theory

number theory unsolved

divisible

Divisibility

Orthocenter lies on circumcircle

by whatshisbucket, Jun 26, 2017, 7:03 AM

Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$

Proposed by Michael Ren

Polish MO Finals 2014, Problem 4

by j___d, Jul 27, 2016, 10:11 PM

Denote the set of positive rational numbers by $\mathbb{Q}_{+}$ . Find all functions $f: \mathbb{Q}_{+}\rightarrow \mathbb{Q}_{+}$ that satisfy
$\underbrace{f(f(f(\dots f(f}_{n}(q))\dots )))=f(nq)$ for all integers $n\ge 1$ and rational numbers $q>0$ .

contests

function

functional equation

Incenter perpendiculars and angle congruences

by math154, Jul 2, 2012, 3:13 AM

$ABC$ is a triangle with incenter $I$ . The foot of the perpendicular from $I$ to $BC$ is $D$ , and the foot of the perpendicular from $I$ to $AD$ is $P$ . Prove that $\angle BPD = \angle DPC$ .

Alex Zhu.

geometry

Submit

Here goes first post of 2025! Great blog.
by math_holmes15, Jan 14, 2025, 8:53 AM
First post of 2024
by Yiyj1, Feb 8, 2024, 5:40 AM
First post of 2023
by HoRI_DA_GRe8, Jul 22, 2023, 7:45 AM
Nice blog ! Your isogonality lemma is really powerful !
by 554183, Oct 14, 2021, 8:55 AM
Post plss....
by samrocksnature, Apr 11, 2021, 10:12 PM
alas,this is ded
by Hamroldt, Mar 18, 2021, 4:13 PM
Thanks for the nice blog.
by Feridimo, Mar 6, 2020, 4:17 PM
I think this might be silly but ... when should we expect to have another post ?? I am very keen to see it
by gamerrk1004, Nov 4, 2019, 4:54 PM
Let's all echo what's written in the blog description - Stay Insane / 'Cause it's your labor, will and pain/ That takes you to the top of soda fountain
by Kayak, Oct 2, 2017, 7:18 PM
hey utkarsh jee is over now ... continue your elementary blog pleaseeeeeee!
by kk108, Jun 17, 2017, 11:19 AM
Congrats on becoming a contest moderator!
by Ankoganit, Mar 9, 2017, 5:22 AM
INTERSTING BLOG
by kk108, Feb 19, 2017, 2:04 PM
I have no plans for this blog right now....
No time here people !
But lets see....
I may try some combinatorics
by utkarshgupta, Feb 15, 2017, 12:47 PM
Thanks for the nice blog!
by Orkhan-Ashraf_2002, Feb 13, 2017, 6:34 PM
Revive it!!!
Best blog out there, for sure!
by rmtf1111, Jan 12, 2017, 6:02 PM
Oh you certainly will..
Bu I don't think I will
by achilles04, Feb 11, 2015, 6:24 PM
Thanks
Hope you are there too ...

And hope I get selected
by utkarshgupta, Feb 11, 2015, 11:14 AM
Nice blog ..
keep it up and you might one day become a mathematician lIke me
( just joking )
Btw don't forget us after going to imotc.
by achilles04, Feb 10, 2015, 4:49 PM
nice blog!!!!!
by pradhit, Feb 7, 2015, 11:51 AM
No ideas about the cut off but should be less than 60 according to me...
So you never know
by utkarshgupta, Feb 6, 2015, 4:06 PM
Hi utkarsh,
what do u expect the cutoff to be this year??

by kgdpsdurg, Feb 6, 2015, 12:41 PM
Hi utkarsh
How many did you clear in INMO 2015?
by moldevort, Feb 1, 2015, 3:59 PM
@ Anonymous Bunny
If we both (that should have included me) are lucky enough !
by utkarshgupta, Sep 17, 2014, 2:44 AM
Nice blog. Great job!

If I am lucky enough, hopefully we shall meet at IMOTC.
by AnonymousBunny, Sep 16, 2014, 8:08 PM
Thanx all !!

If anyone wish to contribute anything just pm !

I will add u to the contributors' list !
by utkarshgupta, Sep 16, 2014, 6:35 AM
doing good progress!! btw nice blog.
by rachitgoel, Sep 15, 2014, 4:21 PM
Nice blog.
by Kezer, Aug 5, 2014, 4:04 PM
Hi Utkarsh! Wonderful blog. Keep it up.
by YESMAths, Jul 20, 2014, 1:36 PM
I need some shouts and characters

by utkarshgupta, Jul 20, 2014, 4:22 AM