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
Thursday at 11:16 PM
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
Thursday at 11:16 PM
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
BMO 2025
GreekIdiot   10
N 3 minutes ago by tranducphat
Does anyone have the problems? They should have finished by now.
10 replies
GreekIdiot
Apr 27, 2025
tranducphat
3 minutes ago
Infinitely many numbers of a given form
Stefan4024   19
N 44 minutes ago by cursed_tangent1434
Source: EGMO 2016 Day 2 Problem 6
Let $S$ be the set of all positive integers $n$ such that $n^4$ has a divisor in the range $n^2 +1, n^2 + 2,...,n^2 + 2n$. Prove that there are infinitely many elements of $S$ of each of the forms $7m, 7m+1, 7m+2, 7m+5, 7m+6$ and no elements of $S$ of the form $7m+3$ and $7m+4$, where $m$ is an integer.
19 replies
Stefan4024
Apr 13, 2016
cursed_tangent1434
44 minutes ago
Very easy case of a folklore polynomial equation
Assassino9931   1
N an hour ago by iamnotgentle
Source: Bulgaria EGMO TST 2025 P6
Determine all polynomials $P(x)$ of odd degree with real coefficients such that $P(x^2 + 2025) = P(x)^2 + 2025$.
1 reply
Assassino9931
2 hours ago
iamnotgentle
an hour ago
Process on scalar products and permutations
Assassino9931   2
N an hour ago by Assassino9931
Source: RMM Shortlist 2024 C1
Fix an integer $n\geq 2$. Consider $2n$ real numbers $a_1,\ldots,a_n$ and $b_1,\ldots, b_n$. Let $S$ be the set of all pairs $(x, y)$ of real numbers for which $M_i = a_ix + b_iy$, $i=1,2,\ldots,n$ are pairwise distinct. For every such pair sort the corresponding values $M_1, M_2, \ldots, M_n$ increasingly and let $M(i)$ be the $i$-th term in the list thus sorted. This denes an index permutation of $1,2,\ldots,n$. Let $N$ be the number of all such permutations, as the pairs run through all of $S$. In terms of $n$, determine the largest value $N$ may achieve over all possible choices of $a_1,\ldots,a_n,b_1,\ldots,b_n$.
2 replies
Assassino9931
3 hours ago
Assassino9931
an hour ago
Square problem
Jackson0423   2
N an hour ago by Jackson0423
Construct a square such that the distances from an interior point to the vertices (in clockwise order) are
1,7,8,4 respectively.
2 replies
Jackson0423
Yesterday at 4:08 PM
Jackson0423
an hour ago
IMO Shortlist Problems
ABCD1728   2
N an hour ago by ABCD1728
Source: IMO official website
Where can I get the official solution for ISL before 2005? The official website only has solutions after 2006. Thanks :)
2 replies
1 viewing
ABCD1728
Yesterday at 12:44 PM
ABCD1728
an hour ago
Estimate on number of progressions
Assassino9931   1
N an hour ago by BlizzardWizard
Source: RMM Shortlist 2024 C4
Let $n$ be a positive integer. For a set $S$ of $n$ real numbers, let $f(S)$ denote the number of increasing arithmetic progressions of length at least two all of whose terms are in $S$. Prove that, if $S$ is a set of $n$ real numbers, then
\[ f(S) \leq \frac{n^2}{4} + f(\{1,2,\ldots,n\})\]
1 reply
Assassino9931
3 hours ago
BlizzardWizard
an hour ago
2^x+3^x = yx^2
truongphatt2668   10
N an hour ago by MittenpunktpointX9
Prove that the following equation has infinite integer solutions:
$$2^x+3^x = yx^2$$
10 replies
truongphatt2668
Apr 22, 2025
MittenpunktpointX9
an hour ago
find the radius of circumcircle!
jennifreind   1
N an hour ago by ricarlos
In $\triangle \rm ABC$, $  \angle \rm B$ is acute, $\rm{\overline{BC}} = 8$, and $\rm{\overline{AC}} = 3\rm{\overline{AB}}$. Let point $\rm D$ be the intersection of the tangent to the circumcircle of $\triangle \rm ABC$ at point $\rm A$ and the perpendicular bisector of segment $\rm{\overline{BC}}$. Given that $\rm{\overline{AD}} = 6$, find the radius of the circumcircle of $\triangle \rm BCD$.
IMAGE
1 reply
jennifreind
Yesterday at 2:12 PM
ricarlos
an hour ago
A folklore polynomial game
Assassino9931   1
N 2 hours ago by YaoAOPS
Source: RMM Shortlist 2024 A1, also Bulgaria Regional Round 2016, Grade 12
Fix a positive integer $d$. Yael and Ziad play a game as follows, involving a monic polynomial of degree $2d$. With Yael going first, they take turns to choose a strictly positive real number as the value of one of the coecients of the polynomial. Once a coefficient is assigned a value, it cannot be chosen again later in the game. So the game
lasts for $2d$ rounds, until Ziad assigns the final coefficient. Yael wins if $P(x) = 0$ for some real
number $x$. Otherwise, Ziad wins. Decide who has the winning strategy.
1 reply
Assassino9931
3 hours ago
YaoAOPS
2 hours ago
Game on board with gcd and lcm
Assassino9931   0
2 hours ago
Source: Bulgaria EGMO TST 2025 P3
On the board are written $n \geq 2$ positive integers with least common multiple $K$ and greatest common divisor $1$. It is known that $K$ is not a perfect square and is not among the initially written numbers. Two players $A$ and $B$ play the following game, taking turns alternatingly, with $A$ starting first. In a move the player has to write a number which has not been written so far, by taking two distinct integers $a$ and $b$ from the board and write LCM$(a,b)$ or LCM$(a,b)$/$a$. The player who writes $1$ or $K$ loses. Who has a winning strategy?
0 replies
Assassino9931
2 hours ago
0 replies
Popular children at camp with algebra and geometry
Assassino9931   0
3 hours ago
Source: RMM Shortlist 2024 C3
Fix an odd integer $n\geq 3$. At a maths camp, there are $n^2$ children, each of whom selects
either algebra or geometry as their favourite topic. At lunch, they sit at $n$ tables, with $n$ children
on each table, and start talking about mathematics. A child is said to be popular if their favourite
topic has a majority at their table. For dinner, the students again sit at $n$ tables, with $n$ children
on each table, such that no two children share a table at both lunch and dinner. Determine the
minimal number of young mathematicians who are popular at both mealtimes. (The minimum is across all sets of topic preferences and seating arrangements.)
0 replies
1 viewing
Assassino9931
3 hours ago
0 replies
Triangles in dissections
Assassino9931   0
3 hours ago
Source: RMM Shortlist 2024 C2
Fix an integer $n\geq 3$ and let $A_1A_2\ldots A_n$ be a convex polygon in the plane. Let $\mathcal{M}$ be the set of all midpoints $M_{i,j}$ of segments $A_iA_j$ where $i\neq j$. Assume that all of these midpoints are distinct, i.e. $\mathcal{M}$ consists of $\frac{n(n-1)}{2}$ elements. Dissect the polygon $M_{1,2}M_{2,3}\ldots M_{n,1}$ into triangles so that the following hold:

(1) The intersection of every two triangles (interior and boundary) is either empty or a common
vertex or a common side.
(2) The vertices of all triangles lie in M (not all points in M are necessarily used).
(3) Each side of every triangle is of the form $M_{i,j}M_{i,k}$ for some pairwise distinct indices $i,j,k$.

Prove that the total number of triangles in such a dissection is $3n-8$.
0 replies
Assassino9931
3 hours ago
0 replies
Tangency geo
Assassino9931   0
3 hours ago
Source: RMM Shortlist 2024 G1
Let $ABC$ be an acute triangle with $\angle ABC > 45^{\circ}$ and $\angle ACB > 45^{\circ}$. Let $M$ be the midpoint of the side $BC$. The circumcircle of triangle $ABM$ intersects the side $AC$ again at $X\neq A$ and the circumcircle of triangle $ACM$ intersects the side $AB$ again at $Y\neq A$. The point $P$ lies on the perpendicular bisector of the segment $BC$ so that the points $P$ and $A$ lie on the same side of $XY$ and $\angle XPY = 90^{\circ} + \angle BAC$. Prove that the circumcircles of triangles $BPY$ and $CPX$ are tangent.
0 replies
Assassino9931
3 hours ago
0 replies
Concurrency
Omid Hatami   13
N Dec 4, 2024 by cursed_tangent1434
Source: Iran TST 2008
Suppose that $ I$ is incenter of triangle $ ABC$ and $ l'$ is a line tangent to the incircle. Let $ l$ be another line such that intersects $ AB,AC,BC$ respectively at $ C',B',A'$. We draw a tangent from $ A'$ to the incircle other than $ BC$, and this line intersects with $ l'$ at $ A_1$. $ B_1,C_1$ are similarly defined. Prove that $ AA_1,BB_1,CC_1$ are concurrent.
13 replies
Omid Hatami
May 20, 2008
cursed_tangent1434
Dec 4, 2024
Concurrency
G H J
Source: Iran TST 2008
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Omid Hatami
1275 posts
#1 • 7 Y
Y by TRYTOSOLVE, itslumi, cadaeibf, Adventure10, Mango247, Mango247, and 1 other user
Suppose that $ I$ is incenter of triangle $ ABC$ and $ l'$ is a line tangent to the incircle. Let $ l$ be another line such that intersects $ AB,AC,BC$ respectively at $ C',B',A'$. We draw a tangent from $ A'$ to the incircle other than $ BC$, and this line intersects with $ l'$ at $ A_1$. $ B_1,C_1$ are similarly defined. Prove that $ AA_1,BB_1,CC_1$ are concurrent.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mathangel
316 posts
#2 • 1 Y
Y by Adventure10
I think that we could reduce the problem to the following:

Given triangle $ ABC$ and $ (I)$ is its incircle. Let $ A', B'$ be points on $ BC, CA$ and $ d$ is a tangent of $ (I)$. Tangents of $ (I)$ from $ A', B'$ (different from $ BC, CA$) intersect $ d$ at $ A_1, B_1$ respectively. Then $ AA_1, BB_1, A'B'$ are concurrent.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Shishkin
99 posts
#3 • 2 Y
Y by Adventure10, Mango247
Lets incircle tachs $ AB$, $ BC$, $ CA$ in $ F$, $ E$, $ D$ respectively.
Points $ F'$, $ E'$, $ D'$ lie on incircle. $ D' \in A_{1}A'$, $ E' \in B_{1}B'$, $ F' \in C_{1}C'$.
$ X = EF \cap LD'$ $ Y = DF \cap LE'$ $ Z = DE \cap LF'$
Let $ P$ be a pole of line $ l$. Then $ P = DD' \cap EE' \cap FF'$
$ AA_{1}$, $ BB_{1}$, $ CC_{1}$ are concurent. $ \Longleftrightarrow$ $ X$, $ Y$, $ Z$ are colinear. ($ X$ is a pole of $ AA_{1}$)
From Pascal teorem for $ DEFF'LD'$ we get that $ X$, $ Y$, $ P$ are colinear. Similarly $ X$, $ Z$, $ P$ are colinear. So $ X$, $ Y$, $ Z$, $ P$ are colinear.
Attachments:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mathangel
316 posts
#4 • 2 Y
Y by Adventure10, ehuseyinyigit
My solution is similar to yours.

Let $ D, E$ be intersections of $ B'B_1, AB$ and $ l, AC$. We easily note that the hexagon $ B'EA_1A'BD$ circumscribes the incircle of triangle $ ABC$. It follows from Brianchon's Theorem that $ A'B', BE, A_1D$ are concurrent at $ K$. Hence, applying Pappus' Theorem to the 2 triples $ (B_1, A_1, E)$ and $ (A, B, D)$, we have the 3 points $ B', K$ and the intersection $ H$ of $ AA_1, BB_1$ are collinear. Hence, $ H$ is on $ A'B'$.
This means $ AA_1, BB_1, A'B'$ are concurrent.

The problem then follows from this.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
thecmd999
2860 posts
#5 • 2 Y
Y by Adventure10, Mango247
Solution
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
#6 • 15 Y
Y by navi_09220114, Eray, Fermat_Theorem, AlastorMoody, amar_04, Kamran011, mijail, SenatorPauline, Nymoldin, parola, mathleticguyyy, Adventure10, Mango247, EpicBird08, and 1 other user
My solution:

From Brianchon theorem ( for $ A'CAC'C_1A_1 $ and $ B'ABA'A_1B_1 $ )
we get $ AA_1, BB_1, CC_1, l $ are concurrent .

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.
sayantanchakraborty
505 posts
#7 • 2 Y
Y by Adventure10, Mango247
Before proving the statement I need a prerequisite lemma:

Lemma:With respect to a given circle $\omega$,the polars are concurrent if and only if their respective poles are collinear.

Proof:Consider the intersection points of the polars and use La Hire's theorem.

Now back to our problem.I shall first name the points:$D,E,F$ are the tangency points of the incircle with $BC,CA,AB$ respectively.Let $X$ be the point on $(I)$ such that $B'X$ is tangent to $(I)$ (other than $E$),$Y$ be the point where $l'$ touches $(I)$,$Z$ be the point where $A'A_1$ touches $(I)$ and $L$ be the point where $C'C_1$ touches $(I)$.In the rest of my proof the poles and polars,etc will be considered with respect to the incircle.

Note that the polars of $A',B',C'$ are $DZ,EX,FL$ and as $A',B',C'$ are collinear,by our lemma $DZ,EX,FL$ concur.Set $YX \cap DF=P,EF \cap YZ=Q,YL \cap DE=R$.Applying Pascal's theorem on the cyclic hexagon $XYZDFE$ we see that $P,Q,EX \cap DZ$ are collinear.Similarly Pascal's theorem in $XYLFDE$ yeilds that $P,R,XE \cap LF$ are collinear.But $XE \cap DZ=XE \cap LF$ as $DZ,EX,FL$ concur so $P,Q,R$ are collinear.

Now see that $DF$ is the polar of $B$,while $XY$ is the polar of $B_1$.Hence by La Hire's theorem $P=DF \cap XY$ is the pole of $BB_1$.Similarly $Q$ is the pole of $AA_1$ and $R$ is the pole of $CC_1$.Finally by our lemma,as $P,Q,R$ are collinear,$AA_1,BB_1,CC_1$ concur,as desired.
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
#8 • 7 Y
Y by GeoMetrix, mueller.25, Hexagrammum16, BinomialMoriarty, Bumblebee60, A-Thought-Of-God, Mathematicsislovely
Wonderful Problem!!! I guess there is a solution with Dual of Desargues Involution too, can anyone provide that solution too. :)
Iran TST 2008 P2 wrote:
Suppose that $ I$ is incenter of triangle $ ABC$ and $ l'$ is a line tangent to the incircle. Let $ l$ be another line such that intersects $ AB,AC,BC$ respectively at $ C',B',A'$. We draw a tangent from $ A'$ to the incircle other than $ BC$, and this line intersects with $ l'$ at $ A_1$. $ B_1,C_1$ are similarly defined. Prove that $ AA_1,BB_1,CC_1$ are concurrent.

Let the Tangency Points made by $\odot(I)$ with $\{BC,C,A,AB\}$ be $\{D,E,F\}$ respectively. Let $\ell'$ be tangent to $\odot(I)$ at a point $X$ andlet $\{A'P,B'Q,C'R\}$ be the other tangents where $\{P,Q,R\}\in\odot(I)$.

Notice that the Polars of $\{A',B',C'\}$ are $\{PD,QE,RF\}$ respectively WRT $\odot(I)$ and as $\overline{A'-B'-C'}$. Hence, $PD,QE,RF$ are concurrent. Now by Pascal on $XQPDFR$ we get that $XQ\cap FD,QP\cap FR,PD\cap RX$ are collinear. Now Pascal on $XPDEFR$ we get that $XP\cap EF,PD\cap FR,DE\cap RX$ are collinear. So combining both the Collinearities we get that $XP\cap EF, DE\cap XR,FD\cap XQ$ are collinear. Now the Polar of $XP\cap EF$ is $AA_1$, the Polar of $XQ\cap FD$ is $BB_1$ and the Polar of $DE\cap RX$ is $CC_1$. So, $AA_1,BB_1,CC_1$ are concurrent.
This post has been edited 3 times. Last edited by amar_04, Mar 6, 2020, 6:25 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
rcorreaa
238 posts
#9
Y by
Suppose that $\ell$ touches the incircle at $P$.

We will prove that $AA_1,B_1,CC_1,\ell$ are concurrent.

Firstly, we will show that $BB_1,CC_1,\ell$ are concurrent, and similarly, we would have that $\{AA_1,CC_1,\ell\}; \{AA_1,BB_1,\ell\}$ are concurrent, implying the desired result.

Now, we use Moving Points. Fix $A_1,B_1,C_1$ and move $P$ projectively on the incircle. Let $T_A=C'C_1 \cap B'B_1$.

$\implies I$ is the incenter of $\Delta C_1T_AB_1$ $\implies \angle ZIY=90º+\frac{\angle FT_AE}{2}$ (fixed).

Hence, the composed map $$C'C_1 \mapsto \mathcal{C}_I \mapsto \mathcal{C}_I \mapsto B'B_1$$given by $$C_1 \mapsto IC_1 \mapsto IB_1 \mapsto B_1$$is projective $(*)$, since it's a projection followed by a rotation with fixed angle through $I$, followed by another projection.

Now, define $CC_1 \cap \ell= Q_C, BB_1 \cap \ell= Q_B$.

Thus, the composed map $$C_1C' \mapsto \mathcal{C}_C \mapsto \mathcal{C}_C \mapsto \ell$$given by $$ A_1 \mapsto CA_1 \mapsto CQ_C \mapsto Q_C$$is projective, since all maps are projections, which are projective maps.

Similarly, $Y \mapsto Q_B$ is projective $\implies$ from $(*)$, the map $$Q_B \mapsto B_1 \mapsto C_1 \mapsto Q_C$$is projective. Hence, since $Q_B,Q_C \in \ell$, which is a fixed line, by the Moving Points Lemma, in order to prove that $Q_B=Q_C$, it is sufficient to verify the problem for $3$ choices of $P$.

Choosing $P$ to be the points such that the incircle touches $AB,BC,CA$, we get the desired result easily.

Hence, we are done.

$\blacksquare$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
starchan
1606 posts
#10
Y by
We will henceforth work in the projective plane. First of all send $\ell'$ to the line at infinity, so that $A', B', C'$ are the points at infinity for $BC, CA, AB$ respectively. Under this transformation, however the incircle becomes an ellipse. In order to fix this; we take an affine homography converting the ellipse into a circle again. Since affine transforms preserve the line at infinity, $\ell'$ remains the line at infinity. Now considering the projective dual of the problem, it can be seen to be equivalent to the following:

Reduced Problem:
Given a triangle $ABC$ with circumcircle $\gamma$, an arbitrary point $P$ on $\gamma$ and $A', B', C'$ as the respective $A, B, C$ antipodes with respect to $\gamma$ prove that $A'P \cap BC, B'P \cap CA$ and $C'P \cap AB$ are collinear.

Now this is easy to prove by spamming Pascal's Hexagram Theorem. Using Pascal on $APCA'BC'$ yields that centre of $\gamma$, $A'P \cap BC$, $C'P \cap AB$ are collinear. Reiterating this result for other pairs of intersections yields that $A'P \cap BC, B'P \cap CA$ and $C'P \cap AB$ are collinear and thus 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.
TheHazard
93 posts
#11 • 3 Y
Y by popop614, dolphinday, ohiorizzler1434
Polar reciprocate the entire diagram to get the equivalent problem statement.
New Problem Statement wrote:
Let $ABC$ be a triangle with point $X$ in the plane and point $P$ on circumcircle $\Gamma$. Let $XA, XB, XC$ intersect $\Gamma$ at $D, E, F$. Then let $DY, EY, FY$ intersect $BC, AC, AB$ at $G, H, I$. Show that $G, H, I$ are collinear.

Now, take a homography in $\mathbb{CP}^2$ that maps $X$ to a point of infinity. Let $L$ be one of the midpoint of arcs $AD, BE, CF$. Oh wait, this is just USAMO 2012/5 with $\gamma = PL$.
This post has been edited 1 time. Last edited by TheHazard, Nov 7, 2023, 6:52 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
OronSH
1730 posts
#12 • 6 Y
Y by centslordm, ihatemath123, GeoKing, ohiorizzler1434, ehuseyinyigit, EpicBird08
brianchon on $ABDXYE,ACDXZF$ gives $AX,BY,CZ,DEF$ concurrent done
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
matematica007
17 posts
#13
Y by
We apply Brianchon on $A_1A'BAB'B_1$ so we obtained that $AA_1,BB_1,l$ are concurrent.
Analogous $BB_1,CC_1,l$ are concurrent so $AA_1,BB_1,CC_1$ are concurrent. So the problem is proved.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
cursed_tangent1434
612 posts
#14
Y by
Denote by $\omega$ the incircle of $\triangle ABC$. We claim that in particular $ AA_1,BB_1,CC_1$ are concurrent on line $\ell$.

Now, note that each side of $AC'C_1A_1A'C$ and $BC'C_1B_1B'C$ are tangent to $\omega$. Thus both these hexagons are tangential, which implies that by Brianchon's Theorem, $AA_1$ and $CC_1$ intersect on $\ell$ and also $BB_1$ and $CC_1$ intersect on $\ell$, which finishes the problem.
Z K Y
N Quick Reply
G
H
=
a