Stay ahead of learning milestones! Enroll in a class over the summer!

G
Topic
First Poster
Last Poster
k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Tuesday, Jun 10 - Aug 26

Calculus
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!


MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
May 1, 2025
0 replies
Geometry
Lukariman   6
N an hour ago by Curious_Droid
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that $\angle HDM$ = 2∠AMP.
6 replies
Lukariman
Yesterday at 12:43 PM
Curious_Droid
an hour ago
Powers of a Prime
numbertheorist17   33
N an hour ago by OronSH
Source: USA TSTST 2014, Problem 6
Suppose we have distinct positive integers $a, b, c, d$, and an odd prime $p$ not dividing any of them, and an integer $M$ such that if one considers the infinite sequence \begin{align*}
		ca &- db \\
		ca^2 &- db^2 \\
		ca^3 &- db^3 \\
		ca^4 &- db^4 \\
&\vdots
	\end{align*} and looks at the highest power of $p$ that divides each of them, these powers are not all zero, and are all at most $M$. Prove that there exists some $T$ (which may depend on $a,b,c,d,p,M$) such that whenever $p$ divides an element of this sequence, the maximum power of $p$ that divides that element is exactly $p^T$.
33 replies
numbertheorist17
Jul 16, 2014
OronSH
an hour ago
Expected Intersections from Random Pairing on a Circle
tom-nowy   2
N an hour ago by lele0305
Let $n$ be a positive integer. Consider $2n$ points on the circumference of a circle.
These points are randomly divided into $n$ pairs, and $n$ line segments are drawn connecting the points in each pair.
Find the expected number of intersection points formed by these segments, assuming no three segments intersect at a single point.
2 replies
tom-nowy
2 hours ago
lele0305
an hour ago
question4
sahadian   5
N 2 hours ago by Mamadi
Source: iran tst 2014 first exam
Find the maximum number of Permutation of set {$1,2,3,...,2014$} such that for every 2 different number $a$ and $b$ in this set at last in one of the permutation
$b$ comes exactly after $a$
5 replies
sahadian
Apr 14, 2014
Mamadi
2 hours ago
No more topics!
Concurrent lines
syk0526   27
N Dec 19, 2024 by ezpotd
Source: North Korea Team Selection Test 2013 #1
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $ BC, CA, AB$ at $ A_1 , B_1 , C_1 $ respectively. The line $AI$ meets the circumcircle of $ABC$ at $A_2 $. The line $B_1 C_1 $ meets the line $BC$ at $A_3 $ and the line $A_2 A_3 $ meets the circumcircle of $ABC$ at $A_4 (\ne A_2 ) $. Define $B_4 , C_4 $ similarly. Prove that the lines $ AA_4 , BB_4 , CC_4 $ are concurrent.
27 replies
syk0526
May 17, 2014
ezpotd
Dec 19, 2024
Concurrent lines
G H J
G H BBookmark kLocked kLocked NReply
Source: North Korea Team Selection Test 2013 #1
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
syk0526
202 posts
#1 • 5 Y
Y by canhhoang30011999, harshmishra, Adventure10, Mango247, GeoKing
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $ BC, CA, AB$ at $ A_1 , B_1 , C_1 $ respectively. The line $AI$ meets the circumcircle of $ABC$ at $A_2 $. The line $B_1 C_1 $ meets the line $BC$ at $A_3 $ and the line $A_2 A_3 $ meets the circumcircle of $ABC$ at $A_4 (\ne A_2 ) $. Define $B_4 , C_4 $ similarly. Prove that the lines $ AA_4 , BB_4 , CC_4 $ are concurrent.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
nima1376
111 posts
#2 • 2 Y
Y by Adventure10, Mango247
$(A_{4},A,B,C)=-1$ $\Rightarrow$ $ AA_4 , BB_4 , CC_4 $ are concurrent in lemone poin of $ABC$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Luis González
4148 posts
#3 • 5 Y
Y by amar_04, Jc426, Adventure10, sabkx, GeoKing
nima1376, you are wrong, the quadrilateral $ABA_4C$ is not harmonic in general. The concurrency point is not the Lemoine point of $\triangle ABC$ but its center $X_{57},$ i.e. the perspector of $\triangle ABC$ and the orthic triangle of $\triangle A_1B_1C_1.$

Let $A_0,B_0,C_0$ be the projections of $A_1,B_1,C_1$ on $B_1C_1,$ $C_1A_1,$ $A_1B_1.$ $D$ and $M$ denote the midpoints of $\overline{BC}$ and the arc $BAC$ of the circumcircle $(O).$

From $(B,C,A_1,A_3)=-1,$ we get $A_3D \cdot A_3A_1=A_3B \cdot A_3C=A_3A_4 \cdot A_3A_2$ $\Longrightarrow$ $DA_1A_4A_2$ is cyclic $\Longrightarrow$ $\angle A_2A_4A_1=\angle A_2DA_1=90^{\circ}$ $\Longrightarrow$ $M \in A_1A_4.$ Furthermore, $A_1A_0 \perp A_3A_0$ implies that $B_1C_1$ bisects $\angle BA_0C$ externally or $\angle BA_0C_1=\angle CA_0B_1.$ Since $\angle BC_1A_0=\angle CB_1A_0,$ then $\triangle BA_0C_1 \sim \triangle CA_0B_1$ $\Longrightarrow$ $C_1A_0:B_1A_0=BC_1:CB_1=BA_1:CA_1,$ therefore isosceles $\triangle AB_1C_1 \sim \triangle MCB$ are similar with corresponding cevians $AA_0$ and $MA_1$ $\Longrightarrow$ $\angle MA_1C=\angle AA_0B_1.$ Hence since $A_0A_1A_4A_3$ is cyclic on account of the right angles at $A_0,A_4,$ we deduce that $A,A_0,A_4$ are collinear and analogously $B_0 \in BB_4$ and $C_0 \in CC_4.$ By Cevian Nest Theorem the conclusion follows.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
nima1376
111 posts
#4 • 2 Y
Y by Adventure10, Mango247
Luis González wrote:
nima1376, you are wrong, the quadrilateral $ABA_4C$ is not harmonic in general. The concurrency point is not the Lemoine point of $\triangle ABC$ but its center $X_{57},$ i.e. the perspector of $\triangle ABC$ and the orthic triangle of $\triangle A_1B_1C_1.$
sorry for my bad mistake
another solution
let $O$ is circumcircle of triangle $ABC$
$A_{2}A_{1}\cap O=K$ .
$(B,C,A_{1},A_{4})=-1\Rightarrow (K,A_{4},B,C)=-1\Rightarrow \frac{BA_{4}}{A_{4}C}=\frac{KB}{KC}$
but $K$ is center of spiral similar goes $BC_{1}$ to $CB_{1}$ $\Rightarrow \frac{BK}{CK}=\frac{BC_{1}}{CB_{1}}$
so we are done with ceva...
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
laFiesta
13 posts
#5 • 8 Y
Y by lebathanh, lminsl, S.C.B., JasperL, amar_04, Adventure10, Mango247, poirasss
North Korea is NOT Korea! "Korea" means South Korea. Why is this in Korea category?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Mikasa
56 posts
#6 • 2 Y
Y by Adventure10, Mango247
First we claim that $\dfrac{A_{4}B}{A_{4}C}=\dfrac{A_{3}B}{A_{3}C}$. For the sake of argument, let us assume that $A_3$ is on the ray $BC$.

Now $\angle BA_{4}A_{3}=180^{\circ}-\dfrac{\angle A}{2}$.
So, $\angle A_{4}A_{3}C=\angle A_{4}A_{3}B=180^{\circ}-\angle BA_{4}A_{3}-\angle A_{4}BA_{3}$
$=\dfrac{\angle A}{2}-\angle A_{4}BA_{3}=\dfrac{\angle A}{2}-\angle A_{4}BC=\angle A_{2}AC-\angle A_{4}AC$
$=\angle A_{2}AA_{4}=\angle A_{2}BA_{4}=\angle A_{2}BA_{4}$.

This means $A_{2}B$ is tangent to the circle $BA_{3}A_{4}$ and $A_{2}C$ is tangent to the circle $CA_{3}A_{4}$. From these information, we have that $A_{4}B=\dfrac{A_{2}B\times A_{3}B}{A_{2}A_{3}}$ and $A_{4}C=\dfrac{A_{2}C\times A_{3}C}{A_{2}A_{3}}$. Since $A_{2}B=A_{2}C$ we have $\dfrac{A_{4}B}{A_{4}C}=\dfrac{A_{3}B}{A_{3}C}$ as we claimed.

By similar arguments, we can show that, $\dfrac{B_{4}C}{B_{4}A}=\dfrac{B_{3}C}{B_{3}A}$ and $\dfrac{C_{4}A}{C_{4}B}=\dfrac{C_{3}A}{C_{3}B}$.

Now apply Menelaus's theorem for the lines $A_{3}B_{1}, B_{3}C_{1}, C_{3}A_{1}$. Then we have three equations:
1) $\dfrac{BA_{3}}{A_{3}C}\cdot \dfrac{CB_{1}}{B_{1}A}\cdot \dfrac{AC_{1}}{C_{1}B}=1$

2) $\dfrac{CB_{3}}{B_{3}A}\cdot \dfrac{AC_{1}}{C_{1}B}\cdot \dfrac{BA_{1}}{A_{1}C}=1$

3) $\dfrac{AC_{3}}{C_{3}B}\cdot \dfrac{BA_{1}}{A_{1}C}\cdot \dfrac{CB_{1}}{B_{1}A}=1$

Multiply (1),(2),(3) and use the fact $AC_1=AB_1, BC_1=BA_1, CA_1=CB_1$ to get that,

$\dfrac{A_{3}B}{A_{3}C}\cdot \dfrac{B_{3}C}{B_{3}A}\cdot \dfrac{C_{3}A}{C_{3}B}=1$

Thus by our previous argument,

4) $\dfrac{A_{4}B}{A_{4}C}\cdot \dfrac{B_{4}C}{B_{4}A}\cdot \dfrac{C_{4}A}{C_{4}B}=1$

Now,by sine law, we have that $\dfrac{A_{4}B}{A_{4}C}=\dfrac{\sin \angle BAA_4}{\sin \angle CAA_4}$. Deriving similar expressions for $\dfrac{B_{4}C}{B_{4}A},\dfrac{C_{4}A}{C_{4}B},$ and using them in (4), we get that ,

$\dfrac{\sin \angle BAA_4}{\sin \angle CAA_4}\cdot \dfrac{\sin \angle CBB_4}{\sin \angle ABB_4}\cdot \dfrac{\sin \angle ACC_4}{\sin \angle BCC_4}=1$.

This the trigonometric form of Ceva's theorem. So $AA_4, BB_4, CC_4$ are concurrent.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
jayme
9792 posts
#7 • 2 Y
Y by Adventure10, Mango247
Dear Mathlinkers,
the problem have something to do with the A-mixtilinear incircle of ABC...
see
http://perso.orange.fr/jl.ayme , A new mixtilinear incircle adventure I, G.G.G. vol. 4, p. 20-21.
Sincerely
Jean-Louis
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Konigsberg
2217 posts
#8 • 2 Y
Y by Adventure10, Mango247
is this too hard for a problem 1 on an olympiad?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
IDMasterz
1412 posts
#9 • 1 Y
Y by Adventure10
To Konigsberg; depends ;) To jayme, yes indeed, my solution shows this directly. Let the perpendicular to $I$ through $AI$ meet $BC$ at $A_5$, then note $A(A_5, A_4; B, C) = -1$. We know that $A_5B_5C_5$ are collinear and perpendicular to $HI$, so $AA_4, BB_4, CC_4$ are concurrent.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
TelvCohl
2312 posts
#10 • 5 Y
Y by amar_04, parola, Adventure10, Clyn, and 1 other user
My solution (for the original problem) :

Since $ AA_1, BB_1, CC_1 $ are concurrent at $ X_{7} $ of $ \triangle ABC $ ,
so from Desargue theorem we get $ A_3 , B_3 , C_3  $ are collinear . ... $ (\star ) $
Since $ A_4A_2, B_4B_2, C_4C_2 $ is the external bisector of $ \angle CA_4B, \angle AB_4C, \angle BC_4A $, respectively ,

so combine with $ ( \star ) $ we get $ \frac{CA_4}{A_4B} \cdot \frac{AB_4}{B_4C} \cdot \frac{BC_4}{C_4A}=\frac{CA_3}{A_3B} \cdot \frac{AB_3}{B_3C} \cdot \frac{BC_3}{C_3A}=1 $ .

ie. $ AA_4, BB_4, CC_4 $ are concurrent

Q.E.D

__________________________________________________
My solution ( for $ AA_4 \cap BB_4 \cap CC_4  \equiv X_{57} $ ) :

Let $ D $ be the projection of $ A_1 $ on $ B_1C_1 $ and $ X $ be the midpoint of arc $ BAC $ .

Since $ (B, C; A_1, A_3)=-1 $ ,

so $ \frac{CA_4}{A_4B}=\frac{CA_3}{A_3B}=\frac{CA_1}{A_1B} $ ,

hence we get $ A_4A_1 $ is the bisector of $ \angle CA_4B $ and $ X \in A_4A_1 $ .

Since $ D(B, C; A_1, A_3)=-1 $ ,
so $ DA_1 $ is the bisector of $ \angle BDC $ ,
hence we get $ \triangle DC_1B \sim \triangle CB_1D $ .

Since $ \frac{B_1D}{DC_1}=\frac{CD}{DB}=\frac{CA_1}{A_1B} $ ,

so we get $ \triangle AC_1B_1 \cap D \sim \triangle XBC \cap A_1 $ ,
hence from $ \angle BAD=\angle BXA_1=\angle BAA_4 $ we get $ D \in AA_4 $ .
ie. $ X_{57} $ of $ \triangle ABC $ lie on $ AA_4 $

Similarly, we can prove $ X_{57} \in BB_4 $ and $ X_{57} \in CC_4 $ ,
so we get $ AA_4, BB_4, CC_4 $ are concurrent at the $ X_{57} $ of $ \triangle ABC $ .

Q.E.D
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
IDMasterz
1412 posts
#11 • 2 Y
Y by Adventure10, Mango247
To prove $AA_4, BB_4, CC_4$ are concurrent at $X_{57}$ from my post, we can do this:

Let $\ell$ be the radical axis of $\odot ABC, \odot A_1B_1C_1$. Let the medial triangle of $A_1B_1C_1$ be $A_6B_6C_6$ and note that $\ell$ is the radical axis of $\odot A_6B_6C_6$ by harmonic conjugates (fact 1). So, note $AA_5 \cap B_1C_1 \in \ell$ for obvious reasons, and because of the fact 1 we conclude $AA_5$ meets $B_1C_1$ at the same point where the orthic axis of $A_1B_1C_1$ meets $B_1C_1$, so $AA_4$ contains the foot of the $A$ altitude of $A_1B_1C_1$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Aiscrim
409 posts
#13 • 2 Y
Y by Adventure10, Mango247
This problem is really weak in the sense that we don't need $A_1,B_1,C_1$ to be the tangency points of the incircle with the sides of $\triangle{ABC}$; it is enough to have $AA_1,BB_1,CC_1$ concurrent.

Let $\{X_A\}=A_1A_2\cap (ABC)$. As $(A_3,A_1,B,C)=-1$, by perspectivity from $A_2$ we get that $X_ABA_4C$ is harmonic. This yields $\dfrac{A_4B}{A_4C}=\dfrac{X_AB}{X_AC}$, but $X_AA_2$ is the bisector of $\widehat{BX_AC}$, whence $\dfrac{A_4B}{A_4C}=\dfrac{A_1B}{A_1C}$.

Writing the analogous relations and multiplying them we get that $$\dfrac{A_4B}{A_4C}\cdot \dfrac{B_4C}{B_4A}\cdot \dfrac{C_4A}{C_4B}= \dfrac{A_1B}{A_1C}\cdot \dfrac{B_1C}{B_1A}\cdot \dfrac{C_1A}{C_1B}=1$$which is equivalent to the fact that $AA_4,BB_4,CC_4$ are concurrent.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Ankoganit
3070 posts
#14 • 2 Y
Y by Adventure10, Mango247
Let $(J_a)$ be a circle tangent to $BC$ at $A_1$ and tangent to $(ABC)$ at some point, say $T_a$. Define $J_b,J_c;T_b,T_c$ similarly. We claim that $A_4\equiv T_a$ etc.

Let $X_a$ denote the midpoint of arc $BAC$ of $(ABC)$. If $\mathcal{H}$ denotes the homothety centered at $T_a$ that takes $(J_a)$ to $(ABC)$, then it takes $BC$ to the line $\ell$ parallel to $BC$ and tangent to $(ABC)$, clearly at $X_a$. Then $\mathcal{H}$ takes $A_1$ to $X_a$; so $T_a,A_1,X_a$ are collinear. Since $X_aA_2$ is a diameter of $(ABC)$, we have $X_aT_a\perp T_aA_2\implies A_1T_a\perp T_aA_2 \;(\star )$.

Next, observe that $(B,C,A_1,A_3)=-1\implies T_a(B,C,A_1,A_3)$ is a harmonic pencil. But $T_aX_a$ and hence $T_aA_1$ is the bisector of $\angle BT_aC$, so we have $A_1T_a\perp T_aA_3$. Combining this with $(\star )$ gives $A_3,T_a,A_2$ are collinear, so $T_a\equiv A_4$.

But from Concurrency on OI we have $AT_a$ etc. are concurrent at the Isogonal Mittenpunkt, as desired. $\blacksquare$
This post has been edited 1 time. Last edited by Ankoganit, May 23, 2016, 5:14 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
lebathanh
464 posts
#15 • 3 Y
Y by Adventure10, Mango247, sami1618
laFiesta wrote:
North Korea is NOT Korea! "Korea" means South Korea. Why is this in Korea category?

I guess they is not good geography :D
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
solver6
259 posts
#16 • 3 Y
Y by alex14rta, Adventure10, Mango247
$\textbf{Proof :}$

Points $A_3, B_3, C_3$ lie on same line $l$. Consider projective transformation which sends circle $(ABC)$ into circle and line $l$ to infinity line. Name $A', B',\ldots, A_2', \ldots , A_4'$ images of points $A, B,\ldots, A_2, \ldots , A_4$ wrt this projective transformation. So we have that $A_2'A_4'|| B'C'$, $B_2'B_4'||A'C'$, $C_2'C_4'||A'B'$. Let lines $A'A_2'$, $B'B_2'$, $C'C_2'$ concurrent at point $X$. Then lines $A'A_4'$, $B'B_4'$, $C'C_4'$ concurrent at point which is isogonal to $X$ wrt $A'B'C'$. $\Box$
This post has been edited 1 time. Last edited by solver6, Mar 1, 2017, 11:54 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
nikolapavlovic
1246 posts
#17 • 2 Y
Y by Adventure10, Mango247
Let $A_2A_1\cap \odot ABC=\{A_5\}$ $(A_5,A_4;B,C)=-1$ $\frac{A_5B}{A_5C}=\frac{CA_1}{BA_1}=\frac{BA_4}{CA_4}$ and hence $BA_4\cdot CB_4\cdot AC_4=A_4C\cdot AB_4 \cdot BC_4$ so we're done.
This post has been edited 2 times. Last edited by nikolapavlovic, Mar 4, 2017, 4:06 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Blast_S1
358 posts
#18 • 1 Y
Y by Adventure10
A pretty bad solution obtained from randomly projecting
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
amar_04
1915 posts
#19 • 4 Y
Y by GeoMetrix, Hexagrammum16, Bumblebee60, mijail
Nice Problem!!!
North Korean TST 2013 P1 wrote:
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $ BC, CA, AB$ at $ A_1 , B_1 , C_1 $ respectively. The line $AI$ meets the circumcircle of $ABC$ at $A_2 $. The line $B_1 C_1 $ meets the line $BC$ at $A_3 $ and the line $A_2 A_3 $ meets the circumcircle of $ABC$ at $A_4 (\ne A_2 ) $. Define $B_4 , C_4 $ similarly. Prove that the lines $ AA_4 , BB_4 , CC_4 $ are concurrent.

Clearly $\{A_2,B_2,C_2\}$ are the midpoints of $\widehat{BC},\widehat{AC},\widehat{AB}$ respectively and let $\{X,Y,Z\}$ be the midpoints of $\widehat{BAC},\widehat{CBA},\widehat{ACB}$ respectively. Then $(A_3,A_1;B,C)\overset{A_4}{=}(A_2,A_1A_4\cap\odot(ABC);B.C)$. So, $A_4A_1\cap\odot(ABC)= X$. Now if $P$ is the foot of perpendicular from $A_1$ to $B_1C_1$ then $AP,XA_1$ concur at $A_4$ from this problem $\longrightarrow$Two Lines meet on a circle. Analogously we get that if $\triangle PQR$ is the orthic triangle of $\triangle A_1B_1C_1$, then $\{\overline{A-P-A_4}\},\{\overline{B-Q-B_4}\},\{\overline{C-R-C_4}\}$. Now as $\{AA_1,BB_1,CC_1\}$ and $\{A_1P.B_1Q,C_1R\}$. Hence by Cevian Nest Theorem we conclude that $AA_4,BB_4,CC_4$ are concurrent. $\blacksquare$
This post has been edited 7 times. Last edited by amar_04, Feb 27, 2020, 5:31 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
william122
1576 posts
#20 • 3 Y
Y by amar_04, crazyeyemoody907, Mango247
Suppose that $A_2A_1$ intersects the circle again at $A'$, and $A_1A_4$ intersects at $A_5$. We have $$(A_3A_1;BC)\stackrel{A_2}{=}(A_4A';BC)\stackrel{A_1}{=}(A_5A_2;BC)=-1$$Thus, $A_4A_5$ is an angle bisector, and $\frac{\sin\angle BAA_4}{\sin\angle CAA_4}=\frac{BA_4}{CA_4}=\frac{BA_1}{A_1C}$. Multiplying similar expressions for $B_4,C_4$, we are done by Trig Ceva and Ceva on the Gergonne point.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
crazyeyemoody907
450 posts
#22
Y by
[asy]
import graph; size(15cm);
real a=105;
real b=205;
real c=335;
real r=10;
path omega=circle(origin, r);
pair A=r*dir(a);
pair B=r*dir(b);
pair C=r*dir(c);
pair I=incenter(A,B,C);
pair A1=foot(I,B,C);
pair B1=foot(I,C,A);
pair C1=foot(I,B,A);
dot(A1);
dot(B1);
dot(C1);
draw(A--B--C--cycle);
draw(omega,blue);
pair X=10*C1-9*B1;
pair P=10*B-9*C;
pair A3=intersectionpoints(B1--X,B--P)[0];
pair A2=r*dir((b+c)/2);
pair B2=r*dir((a+c)/2+180);
pair C2=r*dir((a+b)/2);
dot(A2);
dot(B2);
dot(C2);
pair A4=intersectionpoints(A2--A3,omega)[1];
draw(A2--A3);
dot(A4);
draw(B1--A3);
draw(A3--A2, magenta);
pair D=(B+C)/2;
dot(D);
draw(B--A3);
draw(A1--A4--A2--D--cycle,red+ linewidth(1));
draw(A--A4,green);
label("$A$",A,dir(90));
label("$B$",B,dir(-135));
label("$C$",C,dir(-45));
label("$D$",D,dir(-45));
label("$A_1$",A1,dir(90));
label("$A_2$",A2,dir(-90));
label("$A_3$",A3,dir(-90));
label("$A_4$",A4,dir(-90));
label("$B_1$",B1,dir(45));
label("$C_1$",C1,dir(-45));
draw(B--A4--C,dotted);
label("$B_2$",B2,dir(45));
label("$C_2$",C2,dir(135));
[/asy]
As mentioned earlier, $A_1,B_1,C_1$ may be any points on their respective sides, not necessarily the touch points, as long as $AA_1,BB_1,CC_1$ are concurrent.
First, note that it is well-known that $(A_3A_1;BC)=-1$, and let $D$ be the midpoint of $\overline{BC}$.
By EGMO lemma $9.17$(midpoints of harmonic bundles) combined with PoP, $A_3A_4\cdot A_3A_2=A_3B\cdot A_3C=A_3A_1\cdot A_3D$, thus $A_1A_4A_2D$ is cyclic. Because $\angle A_1DA_2=\pi/2$, this implies $\angle A_1A_4A_2=\pi/2$.
By another well-known property of harmonic bundles, $\overline{A_4A_2}$ bisects $\angle BA_4C$. By angle bisector theorem and law of sines, we have
$\prod_{\text{cyc}}^{}\frac{\sin\angle BAA_4}{\sin\angle A_4AC}=\prod_{\text{cyc}}^{}\frac{2R\sin\angle BAA_4}{2R\sin\angle A_4AC}=\prod_{\text{cyc}}^{}\frac{BA_4}{A_4C}=\prod_{\text{cyc}}^{}\frac{BA_1}{A_1C}$(angles and lengths directed, $R$ is the radius of $(ABC)$)

Because $AA_1,BB_1,CC_1$ concur(at the Gergonne point for the specific case of the problem), the last cyclic product is $1$ by Ceva's theorem.
Because the first cyclic product is 1, by trig Ceva, the lines $AA_4,BB_4,CC_4$ are concurrent, as desired.
This post has been edited 5 times. Last edited by Luis González, Aug 16, 2022, 10:27 PM
Reason: Unhiding solution
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
stonegiraffe247
25 posts
#23
Y by
laFiesta wrote:
North Korea is NOT Korea! "Korea" means South Korea. Why is this in Korea category?

Why? North Korea is still Korea. It encompasses the entire peninsula, should it not?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hakN
429 posts
#24 • 3 Y
Y by Mango247, Mango247, Mango247
Let $D_A = AA_4 \cap BC$ , $S = (AC_1IB_1) \cap (ABC)$ and $E = AS\cap BC$.
It is well-known that $S,A_1,A_2$ are collinear.

Now we have $-1 =(A_3,A_1;B,C) \stackrel{A_2} = (A_4,S;B,C) \stackrel{A} = (D_A,E;B,C)$.

By radical axis theorem on $(ABC) , (BIC) , (ASC_1IB_1)$ we get that $SI$ is tangent to both $(BIC)$ and $(ASC_1IB_1)$.
Hence, we have $\triangle EIB \sim \triangle ECI$.

So, $\frac{EB}{EI} = \frac{EI}{EC} = \frac{IB}{IC} \implies \frac{EB}{EC} = \frac{IB^2}{IC^2}$.
So, $-1 = (D_A,E;B,C) \implies \frac{D_AB}{D_AC} = \frac{EB}{EC} = \frac{IB^2}{IC^2}$.
Writing up similar expressions for the others, we get that $AA_4 , BB_4 , CC_4$ are concurrent by the converse of Ceva's Theorem. $\square$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ike.chen
1162 posts
#25
Y by
By Ceva-Menelaus, we have $-1 = (A_3, A_1; B, C)$. Now, let $A_5$ denote the midpoint of arc $BAC$. Because $$\angle A_3A_4A_5 = 180^{\circ} - \angle A_2A_4A_5 = 90^{\circ}$$and $A_4A_5$ bisects $\angle BA_4C$, the Right Angle and Bisectors Lemma implies $$- 1 = (A_3, A_4A_5 \cap BC ; B, C)$$so $A_1, A_4, A_5$ are collinear.

Let $P, Q, R$ be the feet of the $A_1$-altitude, $B_1$-altitude, $C_1$-altitude respectively of $A_1B_1C_1$. A well-known lemma implies $P \in AA_4, Q \in BB_4, R \in CC_4$. Now, let $H$ denote the orthocenter of $A_1B_1C_1$. Applying the Cevian Nest Lemma on the Gergonne Point of $ABC$ and $H$ implies $AP, BQ, CR$ are concurrent, which finishes. $\blacksquare$


Remark: The well-known lemma I cited is proven in many solutions to TSTST 2020/2.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Tafi_ak
309 posts
#27
Y by
Let $X=(AI)\cap (ABC)$. So by spiral similarity we have $\triangle XBC_1\sim\triangle XCB_1$. So we get $XB/XC=BA_1/CA_1$. Hence $A_1$ lies on the bisector of $\angle BXC$, and that gives $X$, $A_1$, $A_2$ are collinear. So \[ -1=(BC;A_3A_1)\stackrel{A_2}{=}(BC;A_4X)\implies \frac{BX}{CX}=\frac{BA_4}{CA_4} \]Now we have \[ \frac{\sin \angle BAA_4}{\sin \angle CAA_4}=\frac{\sin \angle BCA_4}{\sin \angle CBA_4}=\frac{BA_4}{CA_4}=\frac{BX}{CX}=\frac{BA_1}{CA_1} \]Now by sine ceva we are done.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
WLOGQED1729
44 posts
#28
Y by
Excellent Problem! Kim Jong Un’s math problems never disappoints me. :first:
Step 1 Redefine $A_4,B_4,C_4$
Note that $AA_1,BB_1,CC_1$ are concurrent $\implies$ $(A_3,A_1,B,C)=-1$
Let $M$ be a midpoint of segment $BC$; $A_3A_1\cdot A_3M=A_3B\cdot A_3C=A_3A_4\cdot A_3A_2$.
We conclude that $A_1,M,A_2,A_4$ are concyclic which means $\angle A_1A_4A_2= 90^\circ$.
So, $A_4,A_1,A’$ are collinear where $A’$ is a midpoint of arc $BC$ containing $A$.
Similarly, $B_4,B_1,B’$ and $C_1,C_4,C’$ are collinear where $B’$ is a midpoint of arc $CA$ containing $B$ and $C’$ is a midpoint of arc $AB$ containing $C$.
Step 2 Finish by simple trig-ceva calculation
Observe that $A_1A_4$ bisects $\angle BA_4C$ $\implies \frac{BA_4}{A_4C}=\frac{BA_1}{A_1C}$
Apply law of sines yields $\frac{sin BAA_4}{sin A_4AC}=\frac{BA_4}{A_4C}=\frac{BA_1}{A_1C}$
Apply trigonometric ceva’s theorem to $\triangle ABC$ yields $AA_4, BB_4$ and $CC_4$ are concurrent, as desired.
Attachments:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
pikapika007
298 posts
#29 • 1 Y
Y by GeoKing
bored

In accordance with the absolutely outrageous point naming, let $A_5$ be the $A$-sharkydevil point, and define $B_5$, $C_5$ similarly. Now $A_2$, $A_1$, $A_5$ collinear, so
\[ (A_3, A_1; B, C) \overset{A_2}{=} (A_4, A_5; B, C). \]Thus, if $A_6$ is the foot of the altitude from $A_1$ to $\overline{B_1C_1}$, then $\overline{AA_6A_4}$ collinear (this is well-known), and Cevian Nest finishes.
This post has been edited 1 time. Last edited by pikapika007, Jun 18, 2024, 7:57 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hanulyeongsam
170 posts
#30
Y by
laFiesta wrote:
North Korea is NOT Korea! "Korea" means South Korea. Why is this in Korea category?

DPRK is the only Korea. What do you mean South Korea ????
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ezpotd
1263 posts
#31
Y by
Let $A_5$ be the $A$-Sharkydevil point, or $(AB_1C_1) \cap (ABC)$. Inverting around the incircle yields that $A$ goes to the midpoint of $B_1C_1$ and cyclic variants, implying that $(ABC)$ goes to the nine point circle of $A_1B_1C_1$, since the image of $A_5$ lies on $B_1C_1$ and is not $A$, we know that it is the foot of the altitude from $A_1$ to $B_1C_1$ (calling it $A_6$), giving the result that $A_5, A_6,I$ are collinear. By spiral similarity, we know that $A_5I$ maps to $A_5A_2$, and $B_1C_1$ goes to $BC$, and the $A_6$ goes to $A_1$, since $A_6A_1 \parallel IA_2$ and $A_1$ lies on $B_1C_1$, so $A_5, A_1, A_2$ are collinear. Since $AA_1, BB_1, CC_1$ are concurrenct by Ceva, harmonic lemmas give $(A_3, A_1; B,C)$ harmonic, projecting over $A_2$ gives $(A_4, A_5; B,C)$ harmonic, projecting through $A$ gives $(A_5A \cap B_1C_1, A_4A \cap B_1C_1;B_1,C_1)$ harmonic, projecting through $A_5$ gives $(A, (A_4A \cap B_1C_1)A_5 \cap (AI); B_1, C_1)$ harmonic, so $(A_4A \cap B_1C_1), A_5$ are collinear with $I$, so $A_4A \cap B_1C_1 = A_6$. By Cevian Nest, it is obvious that $AA_6, BB_6, CC_6$, concur, so we are done.
Z K Y
N Quick Reply
G
H
=
a