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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Number Theory Chain!
JetFire008   27
N 8 minutes ago by Maximilian113
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
27 replies
JetFire008
Apr 7, 2025
Maximilian113
8 minutes ago
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ineq.trig.
wer   18
N 9 minutes ago by anduran
If a, b, c are the sides of a triangle, show that: $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\frac{r}{R}\le2$
18 replies
wer
Jul 5, 2014
anduran
9 minutes ago
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P2 Cono Sur 2021
Leo890   9
N 43 minutes ago by jordiejoh
Source: Cono Sur 2021 P2
Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$, respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$, respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$, and the circle $C_2$ intersects $AC$ at $R$ and $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.
9 replies
Leo890
Nov 30, 2021
jordiejoh
43 minutes ago
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collinearity eanted, line tangent to 3 incircles ABP, ACP, BCP related
parmenides51   5
N an hour ago by leon.tyumen
Source: MGO p6 https://artofproblemsolving.com/community/c594864h3379839p31486784
Let $P$ be a point inside $\vartriangle ABC$. It is known that there exists a line tangent to the incircles of $\vartriangle ABP$, $\vartriangle ACP$ and $\vartriangle BCP$. Prove that if $X$ is the intersection point of the common external tangents of a random pair of these incircles and Y is the intersection point of common external tangents of some other pair of these three incircles, then $XY$ passes through either $A$, $B$ or $C$.
5 replies
parmenides51
Sep 2, 2024
leon.tyumen
an hour ago
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Indian RMO - Paper -4[5]
shivangjindal   4
N Feb 23, 2025 by quantam13
In a triangle $ABC$, let $H$ denote its orthocentre. Let $P$ be the reflection of $A$ with respect to $BC$. The circumcircle of triangle $ABP$ intersects the line $BH$ again at $Q$, and the circumcircle of triangle $ACP$ intersects the line $CH$ again at $R$. Prove that $H$ is the incentre of triangle $PQR$.
4 replies
shivangjindal
Dec 12, 2013
quantam13
Feb 23, 2025
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Indian RMO - Paper -4[5]
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Reveal topic tags
geometry
geometric transformation
reflection
circumcircle
incenter
homothety
Reflections
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shivangjindal
676 posts
shivangjindal
#1 h  Dec 12, 2013, 4:52 PM • 2 Y
Y by Adventure10, Mango247
In a triangle $ABC$, let $H$ denote its orthocentre. Let $P$ be the reflection of $A$ with respect to $BC$. The circumcircle of triangle $ABP$ intersects the line $BH$ again at $Q$, and the circumcircle of triangle $ACP$ intersects the line $CH$ again at $R$. Prove that $H$ is the incentre of triangle $PQR$.
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JugHead
217 posts
JugHead
#2 h  Dec 13, 2013, 9:38 AM • 2 Y
Y by Adventure10, Mango247
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AnirrudhRamesh
4 posts
AnirrudhRamesh
#3 h  Dec 18, 2014, 6:17 AM • 1 Y
Y by Adventure10
Triangle PQR is homothetic to orthic triangle of ABC. Done.
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jayme
9775 posts
jayme
#4 h  Feb 9, 2015, 10:49 AM • 1 Y
Y by Adventure10
Dear Mathlinkers,
the homothecy of the two triangles can be proof by considering the circles with diameter the side of ABC and using the Reim's theorem.

Sincerely
Jean-Louis
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quantam13
108 posts
quantam13
#5 h  Feb 23, 2025, 8:50 AM
Y by
It suffices to prove that there is a homothety at $H$ sending the orthic triangle of $\triangle ABC$ to $\triangle PQR$. For this, we need two things, the concurrence of lines $AP$, $BQ$ and $CR$, which is trivial as they are the altitudes and concur at $H$. We also need that line $AB$ is parallel to $PQ$, line $BC$ is parallel to $QR$ and similarly line $RP$ is parallel to $CA$, and this can easily be proven by reim's theorem and the circles with diameters as sides of $\triangle ABC$.
This post has been edited 1 time. Last edited by quantam13, Mar 20, 2025, 2:06 AM
Reason: Clarification
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