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Channel name changed
Plane_geometry_youtuber   10
N an hour ago by Yiyj
Hi,

Due to the search handle issue in youtube. My channel is renamed to Olympiad Geometry Club. And the new link is as following:

https://www.youtube.com/@OlympiadGeometryClub

Recently I introduced the concept of harmonic bundle. I will move on to the conjugate median soon. In the future, I will discuss more than a thousand theorems on plane geometry and hopefully it can help to the students preparing for the Olympiad competition.

Please share this to the people may need it.

Thank you!
10 replies
Plane_geometry_youtuber
Yesterday at 9:31 PM
Yiyj
an hour ago
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Ducks can play games now apparently
MortemEtInteritum   35
N 3 hours ago by pi271828
Source: USA TST(ST) 2020 #1
Let $a$, $b$, $c$ be fixed positive integers. There are $a+b+c$ ducks sitting in a
circle, one behind the other. Each duck picks either rock, paper, or scissors, with $a$ ducks
picking rock, $b$ ducks picking paper, and $c$ ducks picking scissors.
A move consists of an operation of one of the following three forms:

[list]
[*] If a duck picking rock sits behind a duck picking scissors, they switch places.
[*] If a duck picking paper sits behind a duck picking rock, they switch places.
[*] If a duck picking scissors sits behind a duck picking paper, they switch places.
[/list]
Determine, in terms of $a$, $b$, and $c$, the maximum number of moves which could take
place, over all possible initial configurations.
35 replies
MortemEtInteritum
Nov 16, 2020
pi271828
3 hours ago
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2017 IGO Advanced P3
bgn   18
N 3 hours ago by Circumcircle
Source: 4th Iranian Geometry Olympiad (Advanced) P3
Let $O$ be the circumcenter of triangle $ABC$. Line $CO$ intersects the altitude from $A$ at point $K$. Let $P,M$ be the midpoints of $AK$, $AC$ respectively. If $PO$ intersects $BC$ at $Y$, and the circumcircle of triangle $BCM$ meets $AB$ at $X$, prove that $BXOY$ is cyclic.

Proposed by Ali Daeinabi - Hamid Pardazi
18 replies
bgn
Sep 15, 2017
Circumcircle
3 hours ago
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Own made functional equation
JARP091   1
N 3 hours ago by JARP091
Source: Own (Maybe?)
\[ \text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\ f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right) \]
1 reply
JARP091
May 31, 2025
JARP091
3 hours ago
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Euler line of incircle touching points /Reposted/
Eagle116   6
N 4 hours ago by pigeon123
Let $ABC$ be a triangle with incentre $I$ and circumcentre $O$. Let $D,E,F$ be the touchpoints of the incircle with $BC$, $CA$, $AB$ respectively. Prove that $OI$ is the Euler line of $\vartriangle DEF$.
6 replies
Eagle116
Apr 19, 2025
pigeon123
4 hours ago
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Parallel lines on a rhombus
buratinogigle   1
N 4 hours ago by Giabach298
Source: Own, Entrance Exam for Grade 10 Admission, HSGS 2025
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.
1 reply
buratinogigle
5 hours ago
Giabach298
4 hours ago
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Orthocenter lies on circumcircle
whatshisbucket   90
N 4 hours ago by bjump
Source: 2017 ELMO #2
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
90 replies
whatshisbucket
Jun 26, 2017
bjump
4 hours ago
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Polish MO Finals 2014, Problem 4
j___d   3
N 4 hours ago by ariopro1387
Source: Polish MO Finals 2014
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 $q>0$.
3 replies
j___d
Jul 27, 2016
ariopro1387
4 hours ago
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S(an) greater than S(n)
ilovemath0402   1
N 5 hours ago by ilovemath0402
Source: Inspired by an old result
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
1 reply
ilovemath0402
5 hours ago
ilovemath0402
5 hours ago
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Hagge-like circles, Jerabek hyperbola, Lemoine cubic
kosmonauten3114   0
5 hours ago
Source: My own
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 $\triangle{ABC}$.
0 replies
kosmonauten3114
5 hours ago
0 replies
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confusing inequality
giangtruong13   5
N Apr 20, 2025 by arqady
Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{b+c}{a} \geq 2\sqrt{3(ab+bc+ca)}$$
5 replies
giangtruong13
Apr 18, 2025
arqady
Apr 20, 2025
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confusing inequality
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giangtruong13
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giangtruong13
#1 h  Apr 18, 2025, 2:07 PM
Y by
Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{b+c}{a} \geq 2\sqrt{3(ab+bc+ca)}$$
This post has been edited 3 times. Last edited by giangtruong13, Apr 20, 2025, 3:02 PM
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arqady
30263 posts
arqady
#2 h  Apr 18, 2025, 4:14 PM
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giangtruong13 wrote:
Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{a}{b+c} \geq 2\sqrt{3(ab+bc+ca)}$$
It's $$\sum_{cyc}\frac{a}{b+c}\geq2\sqrt{\frac{(ab+ac+bc)(a^2b^2+a^2c^2+b^2c^2)}{a^2b^2c^2}}.$$Are you sure that it's true?
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giangtruong13
153 posts
giangtruong13
#3 h  Apr 19, 2025, 3:03 PM
Y by
Oh sorry, i write wrongly, i will fix it here :oops_sign:
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giangtruong13
153 posts
giangtruong13
#4 h  Apr 20, 2025, 2:41 AM
Y by
Bummppppp
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giangtruong13
153 posts
giangtruong13
#5 h  Apr 20, 2025, 2:57 PM
Y by
This inequality was from a book by an inactive user $toanmuonmau$
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arqady
30263 posts
arqady
#6 h  Apr 20, 2025, 7:57 PM • 1 Y
Y by kiyoras_2001
giangtruong13 wrote:
Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{b+c}{a} \geq 2\sqrt{3(ab+bc+ca)}$$
Because by C-S we obtain:
$$\sum_{cyc}\frac{b+c}{a}=\frac{\sum\limits_{cyc}(a^2b+a^2c)}{abc}=\frac{\sum\limits_{cyc}a^2\sum\limits_{cyc}a-\sum\limits_{cyc}a^3}{abc}=$$$$=\frac{\sqrt{\left(\sum\limits_{cyc}a^2\right)^2\left(\sum\limits_{cyc}a\right)^2}-\sum\limits_{cyc}a^3}{abc}=\frac{\sqrt{\sum\limits_{cyc}(a^4+2a^2b^2)\sum\limits_{cyc}(a^2+2ab)}-\sum\limits_{cyc}a^3}{abc}\geq$$$$\geq\frac{\sqrt{\sum\limits_{cyc}a^4\sum\limits_{cyc}a^2}+2\sqrt{\sum\limits_{cyc}a^2b^2\sum\limits_{cyc}ab}-\sum\limits_{cyc}a^3}{abc}\geq\frac{2\sqrt{\sum\limits_{cyc}a^2b^2\sum\limits_{cyc}ab}}{abc}=2\sqrt{3(ab+ac+bc)}.$$
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