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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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Contests & Programs AMC and other contests, summer programs, etc.
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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.

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0 replies
jlacosta
May 1, 2025
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
Sharygin 2025 CR P12
Gengar_in_Galar   8
N 33 minutes ago by Kappa_Beta_725
Source: Sharygin 2025
Circles $\omega_{1}$ and $\omega_{2}$ are given. Let $M$ be the midpoint of the segment joining their centers, $X$, $Y$ be arbitrary points on $\omega_{1}$, $\omega_{2}$ respectively such that $MX=MY$. Find the locus of the midpoints of segments $XY$.
Proposed by: L Shatunov
8 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
33 minutes ago
Sharygin 2025 CR P17
Gengar_in_Galar   6
N 37 minutes ago by Kappa_Beta_725
Source: Sharygin 2025
Let $O$, $I$ be the circumcenter and the incenter of an acute-angled scalene triangle $ABC$; $D$, $E$, $F$ be the touching points of its excircle with the side $BC$ and the extensions of $AC$, $AB$ respectively. Prove that if the orthocenter of the triangle $DEF$ lies on the circumcircle of $ABC$, then it is symmetric to the midpoint of the arc $BC$ with respect to $OI$.
Proposed by: P.Puchkov,E.Utkin
6 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
37 minutes ago
Sharygin 2025 CR P21
Gengar_in_Galar   4
N an hour ago by Kappa_Beta_725
Source: Sharygin 2025
Let $P$ be a point inside a quadrilateral $ABCD$ such that $\angle APB+\angle CPD=180^{\circ}$. Points $P_{a}$, $P_{b}$, $P_{c},$ $P_{d}$ are isogonally conjugated to $P$ with respect to the triangles $BCD$, $CDA$, $DAB$, $ABC$ respectively. Prove that the diagonals of the quadrilaterals $ABCD$ and $P_{a}P_{b}P_{c}P_{d}$ concur.
Proposed by: G.Galyapin
4 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
an hour ago
Sharygin 2025 CR P18
Gengar_in_Galar   6
N an hour ago by Kappa_Beta_725
Source: Sharygin 2025
Let $ABCD$ be a quadrilateral such that the excircles $\omega_{1}$ and $\omega_{2}$ of triangles $ABC$ and $BCD$ touching their sides $AB$ and $BD$ respectively touch the extension of $BC$ at the same point $P$. The segment $AD$ meets $\omega_{2}$ at point $Q$, and the line $AD$ meets $\omega_{1}$ at $R$ and $S$. Prove that one of angles $RPQ$ and $SPQ$ is right
Proposed by: I.Kukharchuk
6 replies
Gengar_in_Galar
Mar 10, 2025
Kappa_Beta_725
an hour ago
for the contest high achievers, can you share your math path?
HCM2001   28
N an hour ago by N3bula
Hi all
Just wondering if any orz or high scorers on contests at young age (which are a lot of u guys lol) can share what your math path has been like?
- school math: you probably finish calculus in 5th grade or something lol then what do you do for the rest of the school? concurrent enrollment? college class? none (focus on math competitions)?
- what grade did you get honor roll or higher on AMC 8, AMC 10, AIME qual, USAJMO qual, etc?
- besides aops do you use another program to study? (like Mr Math, Alphastar, etc)?

You're all great inspirations and i appreciate the answers.. you all give me a lot of motivation for this math journey. Thanks
28 replies
HCM2001
Wednesday at 7:50 PM
N3bula
an hour ago
Problem 3
EthanWYX2009   5
N an hour ago by parkjungmin
Source: 2023 China Second Round P3
Find the smallest positive integer ${k}$ with the following properties $:{}{}{}{}{}$If each positive integer is arbitrarily colored red or blue${}{}{},$
there may be ${}{}{}{}9$ distinct red positive integers $x_1,x_2,\cdots ,x_9,$ satisfying
$$x_1+x_2+\cdots +x_8<x_9,$$or there are $10{}{}{}{}{}{}$ distinct blue positive integers $y_1,y_2,\cdots ,y_{10}$ satisfiying
$${y_1+y_2+\cdots +y_9<y_{10}}.$$
5 replies
EthanWYX2009
Sep 10, 2023
parkjungmin
an hour ago
Inspired by old results
sqing   1
N 2 hours ago by sqing
Source: Own
Let $a,b,c $ be reals such that $a^2+b^2+c^2=3$ .Prove that
$$(1-a)(k-b)(1-c)+abc\ge -k$$Where $ k\geq 1.$
$$(1-a)(1-b)(1-c)+abc\ge -1$$$$(1-a)(1-b)(1-c)-abc\ge -\frac{1}{2}-\sqrt 2$$
1 reply
sqing
2 hours ago
sqing
2 hours ago
Made from a well-known result
m4thbl3nd3r   0
2 hours ago
1. Let $a,b,c>0$ such that $$\sqrt{(a+b)(a+c)}+\sqrt{(b+a)(b+c)}+\sqrt{(c+a)(c+b)}=3+a+b+c.$$Prove that $$\sqrt{\frac{a+b}{2}}+\sqrt{\frac{b+c}{2}}+\sqrt{\frac{c+a}{2}}\ge ab+bc+ca.$$2. Let $x,y,z$ be sidelengths of a triangle such that $$x^2+y^2+z^2+6=2(xy+yz+zx).$$Prove that $$2\sqrt{2x}+2\sqrt{2y}+2\sqrt{2z}+(x-y)^2+(y-z)^2+(z-x)^2\ge x^2+y^2+z^2.$$
0 replies
m4thbl3nd3r
2 hours ago
0 replies
Interesting inequalities
sqing   5
N 2 hours ago by sqing
Source: Own
Let $ a,b,c,d\geq  0 , a+b+c+d \leq 4.$ Prove that
$$a(kbc+bd+cd)  \leq \frac{64k}{27}$$$$a (b+c) (kb c+  b d+  c d) \leq \frac{27k}{4}$$Where $ k\geq 2. $
5 replies
sqing
Yesterday at 12:44 PM
sqing
2 hours ago
Long and wacky inequality
Royal_mhyasd   5
N 3 hours ago by Royal_mhyasd
Source: Me
Let $x, y, z$ be positive real numbers such that $x^2 + y^2 + z^2 = 12$. Find the minimum value of the following sum :
$$\sum_{cyc}\frac{(x^3+2y)^3}{3x^2yz - 16z - 8yz + 6x^2z}$$knowing that the denominators are positive real numbers.
5 replies
1 viewing
Royal_mhyasd
May 12, 2025
Royal_mhyasd
3 hours ago
Polynomials algebra
Foxellar   2
N 3 hours ago by elizhang101412
\textbf{9.} The real root of the polynomial \( p(x) = 8x^3 - 3x^2 - 3x - 1 \) can be written in the form
\[
\frac{\sqrt[3]{a} + \sqrt[3]{b} + 1}{c},
\]where \( a, b, \) and \( c \) are positive integers. Find the value of \( a + b + c \).
2 replies
Foxellar
5 hours ago
elizhang101412
3 hours ago
4th grader qual JMO
HCM2001   31
N 4 hours ago by Airbus320-214
i mean.. whattttt??? just found out about this.. is he on aops? (i'm sure he is) where are you orz lol..
https://www.mathschool.com/blog/results/celebrating-success-douglas-zhang-is-rsm-s-youngest-usajmo-qualifier
31 replies
HCM2001
Yesterday at 12:53 AM
Airbus320-214
4 hours ago
9 USAMO/JMO
BAM10   23
N 5 hours ago by pieMax2713
I mock ~90-100 on very recent AMC 10 mock right now. I plan to take AMC 10 final fives(9th), intermediate NT(9th), aime A+B courses in 10th and 11th and maybe mathWOOT 1 (12th). For more info I got 20 on this years AMC 8 with 3 sillies and 32 on MATHCOUNTS chapter. Also what is a realistic timeline to do this
23 replies
BAM10
May 19, 2025
pieMax2713
5 hours ago
[TEST RELEASED] OMMC Year 5
DottedCaculator   109
N 5 hours ago by idk12345678
Test portal: https://ommc-test-portal-2025.vercel.app/

Hello to all creative problem solvers,

Do you want to work on a fun, untimed team math competition with amazing questions by MOPpers and IMO & EGMO medalists? $\phantom{You lost the game.}$
Do you want to have a chance to win thousands in cash and raffle prizes (no matter your skill level)?

Check out the fifth annual iteration of the

Online Monmouth Math Competition!

Online Monmouth Math Competition, or OMMC, is a 501c3 accredited nonprofit organization managed by adults, college students, and high schoolers which aims to give talented high school and middle school students an exciting way to develop their skills in mathematics.

Our website: https://www.ommcofficial.org/

This is not a local competition; any student 18 or younger anywhere in the world can attend. We have changed some elements of our contest format, so read carefully and thoroughly. Join our Discord or monitor this thread for updates and test releases.

How hard is it?

We plan to raffle out a TON of prizes over all competitors regardless of performance. So just submit: a few minutes of your time will give you a great chance to win amazing prizes!

How are the problems?

You can check out our past problems and sample problems here:
https://www.ommcofficial.org/sample
https://www.ommcofficial.org/2022-documents
https://www.ommcofficial.org/2023-documents
https://www.ommcofficial.org/ommc-amc

How will the test be held?/How do I sign up?

Solo teams?

Test Policy

Timeline:
Main Round: May 17th - May 24th
Test Portal Released. The Main Round of the contest is held. The Main Round consists of 25 questions that each have a numerical answer. Teams will have the entire time interval to work on the questions. They can submit any time during the interval. Teams are free to edit their submissions before the period ends, even after they submit.

Final Round: May 26th - May 28th
The top placing teams will qualify for this invitational round (5-10 questions). The final round consists of 5-10 proof questions. Teams again will have the entire time interval to work on these questions and can submit their proofs any time during this interval. Teams are free to edit their submissions before the period ends, even after they submit.

Conclusion of Competition: Early June
Solutions will be released, winners announced, and prizes sent out to winners.

Scoring:

Prizes:

I have more questions. Whom do I ask?

We hope for your participation, and good luck!

OMMC staff

OMMC’S 2025 EVENTS ARE SPONSORED BY:

[list]
[*]Nontrivial Fellowship
[*]Citadel
[*]SPARC
[*]Jane Street
[*]And counting!
[/list]
109 replies
DottedCaculator
Apr 26, 2025
idk12345678
5 hours ago
Let's bash some Euler Lines
ABCDE   48
N Sep 2, 2023 by Infinity_Integral
Source: 2014 USAJMO Problem 2
Let $\triangle{ABC}$ be a non-equilateral, acute triangle with $\angle A=60^\circ$, and let $O$ and $H$ denote the circumcenter and orthocenter of $\triangle{ABC}$, respectively.

(a) Prove that line $OH$ intersects both segments $AB$ and $AC$.

(b) Line $OH$ intersects segments $AB$ and $AC$ at $P$ and $Q$, respectively. Denote by $s$ and $t$ the respective areas of triangle $APQ$ and quadrilateral $BPQC$. Determine the range of possible values for $s/t$.
48 replies
ABCDE
Apr 29, 2014
Infinity_Integral
Sep 2, 2023
Let's bash some Euler Lines
G H J
G H BBookmark kLocked kLocked NReply
Source: 2014 USAJMO Problem 2
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ABCDE
1963 posts
#1 • 7 Y
Y by samrocksnature, megarnie, son7, HWenslawski, Adventure10, Mango247, and 1 other user
Let $\triangle{ABC}$ be a non-equilateral, acute triangle with $\angle A=60^\circ$, and let $O$ and $H$ denote the circumcenter and orthocenter of $\triangle{ABC}$, respectively.

(a) Prove that line $OH$ intersects both segments $AB$ and $AC$.

(b) Line $OH$ intersects segments $AB$ and $AC$ at $P$ and $Q$, respectively. Denote by $s$ and $t$ the respective areas of triangle $APQ$ and quadrilateral $BPQC$. Determine the range of possible values for $s/t$.
This post has been edited 1 time. Last edited by djmathman, Apr 6, 2015, 6:43 PM
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AkshajK
4820 posts
#2 • 3 Y
Y by samrocksnature, HWenslawski, Adventure10
Outline
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mymathboy
163 posts
#3 • 3 Y
Y by samrocksnature, HWenslawski, Adventure10
Here is a graph.
Click to reveal hidden text
Attachments:
This post has been edited 6 times. Last edited by mymathboy, Apr 30, 2014, 3:45 AM
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ABCDE
1963 posts
#4 • 4 Y
Y by samrocksnature, HWenslawski, Adventure10, Mango247
2a - The diameter can't go through arc BC because then it doesn't pass through the orthocenter. But wait, it's acute so you can't fit it in one arc so it passes through both and you're done.

2b - I don't think you can do this without trig, but if D is the midpoint of arc BC you get APDQ is a rhombus with one diagonal being 2Rcos something and height of ABC is something R cos something so then everything cancels out nicely and you get a rational linear function which is very easy to find the range of so hoo rah.

Anyone got a "nice" bary, complex, or calc solution?
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DaChickenInc
418 posts
#5 • 4 Y
Y by samrocksnature, HWenslawski, Adventure10, Mango247
Synthetic Solution Sketch
This is a condensation of my fairly rigorous proof.
EDIT: Oops, then I wasted the whole JMO. I was just thinking, "$\angle BOC=120^\circ$ doesn't uniquely define $O$" but thanks to your explanation it turned out to be sufficient to solve the problem. Hopefully I will be more productive on my birthday.
This post has been edited 2 times. Last edited by DaChickenInc, Apr 30, 2014, 12:06 AM
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ABCDE
1963 posts
#6 • 4 Y
Y by samrocksnature, HWenslawski, Adventure10, Mango247
uhh how $BOC=BHC=120$ since $A=60$...?
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numbersandnumbers
258 posts
#7 • 4 Y
Y by samrocksnature, HWenslawski, Adventure10, Mango247
Maybe Nice Solution
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zacchro
179 posts
#8 • 4 Y
Y by samrocksnature, HWenslawski, Adventure10, Mango247
ABCDE wrote:
2b - I don't think you can do this without trig
I got pretty close by continuing coordinate bashing
set point A to be (0,0), C=(1,0), and b=(b,b*sqrt3).
solve for slope of OH=-sqrt3 so APQ is equilateral with area s=(2b+1)^2*sqrt3/36. s/t is then (2b+1)^2/(18b-(2b+1)^2). I couldn't prove that it had range (4/5,1), unfortunately, although I think I could've calculus bashed it to find b=1/2 is the only local minimum.
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mymathboy
163 posts
#9 • 4 Y
Y by samrocksnature, HWenslawski, Adventure10, Mango247
ABCDE wrote:
uhh how $BOC=BHC=120$ since $A=60$...?

$\angle BOC = \angle OBA + \angle BAO + \angle OCA + \angle OAC = 2(\angle BAO + \angle OAC) $
$= 2A = 120^o$

$\angle BHC = 180^o - A = 120^o$
This post has been edited 3 times. Last edited by mymathboy, Apr 30, 2014, 2:56 AM
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mathtastic
3258 posts
#10 • 4 Y
Y by samrocksnature, HWenslawski, Adventure10, Mango247
It should say $60^{\circ}$

How can you prove 2b? I got the answer, but how??
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ABCDE
1963 posts
#11 • 4 Y
Y by samrocksnature, HWenslawski, Adventure10, Mango247
You can WLOG it and scale some length to 1 for convenience and pick some angle or length $l$ to vary (some work out better than others), but in the end $s/t$ becomes some function $f(l)$ and you know what domain you need so you get the range.

I personally set the circumradius as 1 and varied $60-B$ and it worked out very nicely.
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tastymath75025
3223 posts
#12 • 4 Y
Y by samrocksnature, HWenslawski, Adventure10, Mango247
So for part a I coordinate bashed.
Then just now, I read Geo Revisited to look at special properties of Euler lines and I see this:
If ABC has the special property that the Euler line is parallel to side BC, then $\tan B \tan C = 3$.

... Great timing -_-

It basically instant-kills the problem.
One angle is 60, $\tan 60 = \sqrt{3}$, so therefore the tangent of the other angle is $\sqrt{3}$, so it is 60 degrees.
However, the triangle is not equilateral, so contradiction.
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qwerty137
3575 posts
#13 • 4 Y
Y by samrocksnature, HWenslawski, Adventure10, Mango247
Somewhat bad way for 2b: (close to) equilateral, then (close to) 30-60-90 right, then assume those are the boundaries
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matholympiad25
426 posts
#14 • 4 Y
Y by samrocksnature, HWenslawski, Adventure10, Mango247
Is it OK if I proved that, according to mathboy's diagram, $\angle AOH < \angle AOB$ and $180^{\circ} - \angle AOH < \angle AOC$ for part 2a?
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ABCDE
1963 posts
#15 • 4 Y
Y by samrocksnature, HWenslawski, Adventure10, Mango247
vincenthuang75025 wrote:
So for part a I coordinate bashed.
Then just now, I read Geo Revisited to look at special properties of Euler lines and I see this:
If ABC has the special property that the Euler line is parallel to side BC, then $\tan B \tan C = 3$.

... Great timing -_-

It basically instant-kills the problem.
One angle is 60, $\tan 60 = \sqrt{3}$, so therefore the tangent of the other angle is $\sqrt{3}$, so it is 60 degrees.
However, the triangle is not equilateral, so contradiction.


Does it? It's segments AB and AC not lines.
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