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Heavy config geo involving mixtilinear
Assassino9931   1
N 6 minutes ago by VicKmath7
Source: Bulgaria Spring Mathematical Competition 2025 12.4
Let $ABC$ be an acute-angled triangle \( ABC \) with \( AC > BC \) and incenter \( I \). Let \( \omega \) be the mixtilinear circle at vertex \( C \), i.e. the circle internally tangent to the circumcircle of \( \triangle ABC \) and also tangent to lines \( AC \) and \( BC \). A circle \( \Gamma \) passes through points \( A \) and \( B \) and is tangent to \( \omega \) at point \( T \), with \( C \notin \Gamma \) and \( I \) being inside \( \triangle ATB \). Prove that:
$$\angle CTB + \angle ATI = 180^\circ + \angle BAI - \angle ABI.$$
1 reply
Assassino9931
2 hours ago
VicKmath7
6 minutes ago
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Math Competitions
anishka14   5
N Today at 4:33 AM by iwillregretthisnamelater
Hi everyone!

So I am currently in grade 6, and if anyone could give any tips for getting high scores in math competition, that would be great!

I haven't been doing so well in AMC 8, and other competitions like Math Kangaroo, etc....

I feel like i'm stuck, so if anyone could give any resources that helped you learn and score better, could you share that with me?

Thank you so much!

( also how much time should i spend on math every day? )
5 replies
anishka14
Yesterday at 7:55 PM
iwillregretthisnamelater
Today at 4:33 AM
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Something Horrible-a Challenge
Xueshuxue   19
N Today at 3:09 AM by pieMax2713
Hello, I was wondering if it's possible to make 8 with the numbers 5, 3, 5, and 7 under the following rules:
-You can only use 5 twice, 3 once, and 7 once.
-You must use all the numbers.
You can stack numbers to form larger numbers (example: I could take 3 and 5 and turn it into 35 or 53, or use 7, 3, and 5 to make 375.)
-You are allowed to use parentheses.
(Also, I already found out that no 3 digital numbers will work for the solution.)
19 replies
Xueshuxue
Friday at 7:09 PM
pieMax2713
Today at 3:09 AM
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The daily problem!
Leeoz   48
N Today at 3:04 AM by pieMax2713
Every day, I will try to post a new problem for you all to solve! If you want to post a daily problem, you can! :)

Please hide solutions and answers, hints are fine though! :)

The first problem is:
[quote=March 21st Problem]Alice flips a fair coin until she gets 2 heads in a row, or a tail and then a head. What is the probability that she stopped after 2 heads in a row? Express your answer as a common fraction.[/quote]

Past Problems!
48 replies
Leeoz
Mar 21, 2025
pieMax2713
Today at 3:04 AM
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I cannot into nationals, what should I do?
AMathCountsguy10   101
N Today at 2:41 AM by hashbrown2009
I am an eighth grader in North Carolina and I am sort of pressured to make nats this year, I only got 26 on last year's state test (orz to @mathprodigy2011) and I have mocked 39, 40, and 37 on the 2014, 2015, and 2022 tests respectively. I am aiming for a 40 this year because that seems to be the cutoff. How should I sufficiently improve? My test is on March 14th, 2025.

To be more clear, last year I got 16/10 and on the tests I have mocked, I got 27/12, 26/14, and 25/12.
101 replies
AMathCountsguy10
Feb 21, 2025
hashbrown2009
Today at 2:41 AM
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How to pronounce Rusczyk
Dream9   2
N Today at 2:13 AM by KF329
I've always said like ruzz icks. I don't know why I say it as plural but does anyone know how to say the sigma's last name?
2 replies
Dream9
Yesterday at 11:48 PM
KF329
Today at 2:13 AM
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How much does math help?
Dream9   5
N Today at 1:21 AM by yaxuan
Does the math many people on AOPS work for really make a large impact? I get that this is a place for math but I think I'm losing touch with my prior math skill. AMC and AIME are solving problems that have answers and just take practice but math as a job or just like an accountant doesn't need/needs a lot more than this type of math. I might switch to another stem category for college apps pathway. Also, many people who are already very good at competition math don't get good offers because there's people who are even better(note: i get the point of math isn't only college). I think if I work *hard* I'll be AIME qual but I think many people would have higher achievements with the same amount of work in another category. I think competition math really just trains for problem solving, unique thinking, and pattern recognition(aka IQ pts).
5 replies
Dream9
Today at 12:49 AM
yaxuan
Today at 1:21 AM
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problemo
hashbrown2009   1
N Today at 1:03 AM by Dream9
if x/(3^3+4^3) + y/(3^3+6^3) =1

and

x/(5^3+4^3) + y/(5^3+6^3) =1

find the 2 values of x and y.
1 reply
hashbrown2009
Today at 12:49 AM
Dream9
Today at 1:03 AM
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Mathcounts State Sprint Scores
kp34912   31
N Today at 12:42 AM by hashbrown2009
I am practicing previous years' MathCount papers, but they seem to get progressively harder each year.

So, for those of you who scored well in Mathcounts, how many sprint questions are/were you able to do in State/Nationals?

I want to get a sense of where I stand.

Thanks in advance!
31 replies
1 viewing
kp34912
Mar 27, 2025
hashbrown2009
Today at 12:42 AM
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Bogus Proof Marathon
pifinity   7533
N Today at 12:30 AM by Inaaya
Hi!
I'd like to introduce the Bogus Proof Marathon.

In this marathon, simply post a bogus proof that is middle-school level and the next person will find the error. You don't have to post the real solution :P

Use classic Marathon format: 
[hide=P#]a1b2c3[/hide]
[hide=S#]a1b2c3[/hide]


Example posts:

P(x)
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S(x)
P(x+1)
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Let's go!! Just don't make it too hard!
7533 replies
pifinity
Mar 12, 2018
Inaaya
Today at 12:30 AM
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A retirement to be celebrated...
littlebigbull   23
N Yesterday at 11:37 PM by joyjoyy1
Hello Middle School Math!
In case you have not seen the post in the global announcements, Richard Rusczyk is retiring his position as the founder of AOPS!
I would like to respect him in this thread because without him, there would not be a friendly place to chat about math problems, and make friends!

Thank you!


----littlebigbull
23 replies
littlebigbull
Mar 25, 2025
joyjoyy1
Yesterday at 11:37 PM
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Diagonals of a Trapezium
DumSpiroSpero   2
N Jul 10, 2016 by Virgil Nicula
Source: http://mathworld.wolfram.com/Trapezoid.html
In a trapezium with bases $a$ and $b$ and lateral sides $c$ and $d$, proof that the diagonals $p$ and $q$ are given by the following expressions, respectively:

IMAGE

IMAGE
2 replies
DumSpiroSpero
Jul 9, 2016
Virgil Nicula
Jul 10, 2016
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Diagonals of a Trapezium
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Reveal topic tags
geometry
trapezoid
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Source: http://mathworld.wolfram.com/Trapezoid.html
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DumSpiroSpero
68 posts
DumSpiroSpero
#1 h  Jul 9, 2016, 3:13 PM • 1 Y
Y by Adventure10
In a trapezium with bases $a$ and $b$ and lateral sides $c$ and $d$, proof that the diagonals $p$ and $q$ are given by the following expressions, respectively:

http://i.imgur.com/DtAZitz.gif

http://i.imgur.com/Z6qQDAn.gif
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george_54
1585 posts
george_54
#2 h  Jul 9, 2016, 4:50 PM • 2 Y
Y by Adventure10, Mango247
Applying Euler's theorem:
${a^2} + {b^2} + {c^2} + {d^2} = {p^2} + {q^2} + 4M{N^2} = {p^2} + {q^2} + 4\frac{{{{(b - a)}^2}}}{4} \Leftrightarrow $ $\boxed{{p^2} + {q^2} = {c^2} + {d^2} + 2ab}$ $(1)$

Cosine Law:
$\left\{ \begin{array}{l} {p^2} = {b^2} + {c^2} - 2bc\cos D\\ {q^2} = {a^2} + {c^2} - 2ac\cos A \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} a{p^2} = a{b^2} + a{c^2} + 2abc\cos A\\ b{q^2} = {a^2}b + b{c^2} - 2abc\cos A \end{array} \right. \Leftrightarrow $ $\boxed{a{p^2} + b{q^2} = a{b^2} + a{c^2} + {a^2}b + b{c^2}}$ $(2)$

From $(1), (2)$ we get the desired result.
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This post has been edited 1 time. Last edited by george_54, Jul 9, 2016, 4:59 PM
Reason: typo
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Virgil Nicula
7054 posts
Virgil Nicula
#3 h  Jul 10, 2016, 2:20 AM • 2 Y
Y by DumSpiroSpero, Adventure10
See P1 from here
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