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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 2025
0 replies
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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
J
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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
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Solve All 6 IMO 2024 Problems (42/42), New Framework Looking for Feedback
Blackhole.LightKing   4
N 3 minutes ago by Blackhole.LightKing
Hi everyone,

I’ve been experimenting with a different way of approaching mathematical problem solving — a framework that emphasizes recursive structures and symbolic alignment rather than conventional step-by-step strategies.

Using this method, I recently attempted all six problems from IMO 2024 and was able to arrive at what I believe are valid full-mark solutions across the board (42/42 total score, by standard grading).

However, I don’t come from a formal competition background, so I’m sure there are gaps in clarity, communication, or even logic that I’m not fully aware of.

If anyone here is willing to take a look and provide feedback, I’d appreciate it — especially regarding:

The correctness and completeness of the proofs

Suggestions on how to make the ideas clearer or more elegant

Whether this approach has any broader potential or known parallels

I'm here to learn more and improve the presentation and thinking behind the work.

You can download the Solution here.

https://agi-origin.com/assets/pdf/AGI-Origin_IMO_2024_Solution.pdf


Thanks in advance,
— BlackholeLight0


4 replies
Blackhole.LightKing
Yesterday at 12:14 PM
Blackhole.LightKing
3 minutes ago
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$KH$, $EM$ and $BC$ are concurrent
yunxiu   44
N 28 minutes ago by alexanderchew
Source: 2012 European Girls’ Mathematical Olympiad P7
Let $ABC$ be an acute-angled triangle with circumcircle $\Gamma$ and orthocentre $H$. Let $K$ be a point of $\Gamma$ on the other side of $BC$ from $A$. Let $L$ be the reflection of $K$ in the line $AB$, and let $M$ be the reflection of $K$ in the line $BC$. Let $E$ be the second point of intersection of $\Gamma $ with the circumcircle of triangle $BLM$.
Show that the lines $KH$, $EM$ and $BC$ are concurrent. (The orthocentre of a triangle is the point on all three of its altitudes.)

Luxembourg (Pierre Haas)
44 replies
yunxiu
Apr 13, 2012
alexanderchew
28 minutes ago
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An fe based off of another trivial problem
benjaminchew13   1
N 35 minutes ago by Mathzeus1024
Source: Own
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$, $f(x+y-f(x))f(f(x+y)-y)=f(xy)$
1 reply
benjaminchew13
3 hours ago
Mathzeus1024
35 minutes ago
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functional equation interesting
skellyrah   6
N an hour ago by skellyrah
find all functions IR->IR such that $$xf(x+yf(xy)) + f(f(y)) = f(xf(y))^2 + (x+1)f(x)$$
6 replies
skellyrah
Yesterday at 8:32 PM
skellyrah
an hour ago
J
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4 variables with quadrilateral sides
mihaig   1
N an hour ago by Quantum-Phantom
Source: VL
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$4\left(abc+abd+acd+bcd\right)\geq3\left(a+b+c+d\right)+4.$$
1 reply
+1 w
mihaig
5 hours ago
Quantum-Phantom
an hour ago
J
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4 var inequality
sealight2107   1
N an hour ago by sealight2107
Source: Own
Let $a,b,c,d$ be positive reals such that $a+b+c+d+\frac{1}{abcd} = 18$. Find the minimum and maximum value of $a,b,c,d$
1 reply
sealight2107
Wednesday at 2:40 PM
sealight2107
an hour ago
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Inspired by SXTX (4)2025 Q712
sqing   1
N 2 hours ago by sqing
Source: Own
Let $ a ,b,c>0 $ and $ (a+b)^2+2(b+c)^2+(c+a)^2=12. $ Prove that$$ abc(a+b+c) \leq \frac{9}{5} $$Let $ a ,b,c>0 $ and $ 2(a+b)^2+ (b+c)^2+2(c+a)^2=12. $ Prove that$$ abc(a+b+c) \leq \frac{9}{8} $$
1 reply
sqing
Yesterday at 11:59 AM
sqing
2 hours ago
J
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2^x+3^x = yx^2
truongphatt2668   4
N 2 hours ago by Jackson0423
Prove that the following equation has infinite integer solutions:
$$2^x+3^x = yx^2$$
4 replies
truongphatt2668
Apr 22, 2025
Jackson0423
2 hours ago
J
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Operations on Pebbles
MarkBcc168   23
N 2 hours ago by quantam13
Source: ISL 2022 C6
Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.
23 replies
MarkBcc168
Jul 9, 2023
quantam13
2 hours ago
J
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Polynomials
P162008   1
N 2 hours ago by thehound
Define a family of polynomials by $P_{0}(x) = x - 2$ and $P_{k}(x) = \left(P_{k - 1} (x)\right)^2 - 2$ if $k \geq 1$ then find the coefficient of $x^2$ in $P_{k}(x)$ in terms of $k.$
1 reply
P162008
Today at 2:05 AM
thehound
2 hours ago
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Irrational equation
giangtruong13   4
N 2 hours ago by giangtruong13
Solve the equation : $$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$
4 replies
giangtruong13
Yesterday at 1:44 PM
giangtruong13
2 hours ago
J
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\frac{1}{9}+\frac{1}{\sqrt{3}}\geq a^2+\sqrt{a+ b^2} \geq \frac{1}{4}
sqing   3
N 2 hours ago by sqing
Source: Own
Let $a,b\geq 0 $ and $3a+4b =1 .$ Prove that
$$\frac{2}{3}\geq a +\sqrt{a^2+ 4b^2}\geq \frac{6}{13}$$$$\frac{1}{9}+\frac{1}{\sqrt{3}}\geq a^2+\sqrt{a+ b^2} \geq \frac{1}{4}$$$$2\geq a+\sqrt{a^2+16b} \geq \frac{2}{3}\geq a+\sqrt{a^2+16b^3} \geq \frac{2(725-8\sqrt{259})}{729}$$
3 replies
sqing
Oct 3, 2023
sqing
2 hours ago
J
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Find all possible values of BT/BM
va2010   54
N 2 hours ago by ja.
Source: 2015 ISL G4
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
54 replies
va2010
Jul 7, 2016
ja.
2 hours ago
J
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Functions
Potla   23
N 2 hours ago by Ilikeminecraft
Source: 0
Find all functions $ f: \mathbb{R}\longrightarrow \mathbb{R}$ such that
\[f(x+y)+f(y+z)+f(z+x)\ge 3f(x+2y+3z)\]
for all $x, y, z \in \mathbb R$.
23 replies
Potla
Feb 21, 2009
Ilikeminecraft
2 hours ago
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J
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Find the circumradius
Rushil   4
N Aug 23, 2023 by Krishijivi
Source: Indian RMO 1996 Problem 1
The sides of a triangle are three consecutive integers and its inradius is $4$. Find the circumradius.
4 replies
Rushil
Oct 26, 2005
Krishijivi
Aug 23, 2023
J
Find the circumradius
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Reveal topic tags
geometry
circumcircle
inradius
perimeter
L
G H BBookmark kLocked kLocked NReply
Source: Indian RMO 1996 Problem 1
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Rushil
1592 posts
Rushil
#1 h  Oct 26, 2005, 2:52 AM • 1 Y
Y by Adventure10
The sides of a triangle are three consecutive integers and its inradius is $4$. Find the circumradius.
Z K Y
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Shogia
107 posts
Shogia
#2 h  Oct 26, 2005, 3:46 AM • 4 Y
Y by bhutraaditya, Chandrachur, Adventure10, Mango247
Let the sides of the triangle be n-1, n, n+1. This means the semi perimeter of the triangle is 3n/2.
Area=rs=abc/4R
Therefore we have $4(3n/2)=(n^3-n)/(4R)$ which simplifys to $24R=n^2-1$.
Now comparing rs to Hero's Formula we have $4(3n/2)=sqrt((3n/2)(n/2)(n/2+1)(n/2-1))$
pulling out then $n/2$ and canceling gives $12=sqrt(3((n^2/4)-1))$
Squaring both sides results in $144=(3/4)n^2-3$
Multiplying both sides by 4/3 and adding 3 to both sides we obtain $195=n^2-1$

Now we can substitute this into our expression for R and we see that $24R=195$ so the circumradius is $65/8$
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madhusudan kale
23 posts
madhusudan kale
#3 h  May 12, 2012, 10:08 AM • 3 Y
Y by Adventure10, Mango247, namanrobin08
Repeated question RMO-1994 #2.
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sunken rock
4384 posts
sunken rock
#4 h  May 14, 2012, 4:46 PM • 2 Y
Y by Adventure10, Mango247
See here.

Best regards,
sunken rock
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Krishijivi
99 posts
Krishijivi
#5 h  Aug 23, 2023, 6:00 PM
Y by
Let the sides be x, x+1,x+2.
r=4
Area=rs=4(3x+3)/2=6(x+1)
Area=√s(s-x){s- x+1}{s+ x+2}
=√3(x+1)²(x-1)(x+3)/ 16
6(x+1)={√3(x-1)(x+3) }*(x+1)/4
3(x+3)(x-1)=24²
(x-13)(x+15)=0
x≠15
So, x=13
Area=ABC/4R
R=13*14*15/6(13+1)=65/8(Ans)
@Krishijivi
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