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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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
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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   3
N 18 minutes ago by DottedCaculator
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


3 replies
+2 w
Blackhole.LightKing
3 hours ago
DottedCaculator
18 minutes ago
J
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2 var inequalities
sqing   3
N 21 minutes ago by sqing
Source: Own
Let $ a,b> 0 $ and $ a+b\leq 2ab . $ Prove that
$$ \frac{ a + b }{ a^2(1+ b^2)} \leq \sqrt 5-1$$$$ \frac{ a +ab+ b }{ a^2(1+ b^2)} \leq \frac{3(\sqrt5-1)}{2}$$$$ \frac{ a +a^2b^2+ b }{ a^2(1+ b^2)} \leq2$$Solution:
$a\ge\frac{b}{2b-1}, b>\frac12$ and $ \frac{ a +a^2b^2+ b }{ a^2(1+ b^2)} \le\frac{2ab+a^2b^2}{a^2(1+b^2)}=1+\frac{2b-a}{a(1+b^2)} \le 1+\frac{4b-3}{b^2+1}$

Assume $u=4b-3>0$ then $ \frac{ a +a^2b^2+ b }{ a^2(1+ b^2)} \le 1+\frac{16u}{u^2+6u+25} =2+ \frac{16}{6+u+\frac{25}u} \le 3$
Equalityholds when $a=\frac{2}{3},b=2. $
3 replies
sqing
Yesterday at 1:13 PM
sqing
21 minutes ago
J
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hard problem
Cobedangiu   8
N 21 minutes ago by ReticulatedPython
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$
8 replies
Cobedangiu
Apr 21, 2025
ReticulatedPython
21 minutes ago
J
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9 Fun Proof Endings
elasticwealth   23
N 23 minutes ago by drhong
It seems like AOPS is going through a stressful phase right now.

Let's lighten the mood by voting on the best proof endings!
23 replies
+1 w
elasticwealth
Today at 12:26 AM
drhong
23 minutes ago
J
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Irrational equation
giangtruong13   3
N 26 minutes ago by navier3072
Solve the equation : $$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$
3 replies
giangtruong13
2 hours ago
navier3072
26 minutes ago
J
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2 var inequalities
sqing   0
31 minutes ago
Source: Own
Let $ a,b> 0 $ and $ a+b\leq 3ab . $ Prove that
$$ \frac{ a + b }{ a^2(1+ 3b^2)} \leq \frac{3}{2}$$$$ \frac{ a - ab+ b }{ a^2(1+ 3b^2)} \leq 1$$$$ \frac{ a + 3ab+ b }{ a^2(1+ 3b^2)} \leq 3$$$$ \frac{ a -2ab+ b }{ a^2(1+ b^2)}\leq \sqrt{\frac{5}{2}}-\frac{1}{2}$$$$ \frac{ a +ab+ b }{ a^2(1+ b^2)} \leq 2(\sqrt{10}-1)$$$$ \frac{ a -2a^2b^2+ b }{ a^2(1+ b^2)}\leq \frac{\sqrt{82}-5}{2}$$
0 replies
sqing
31 minutes ago
0 replies
J
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Non-negative real variables inequality
KhuongTrang   0
32 minutes ago
Source: own
Problem. Let $a,b,c\ge 0: ab+bc+ca>0.$ Prove that$$\color{blue}{\frac{\left(2ab+ca+cb\right)^{2}}{a^{2}+4ab+b^{2}}+\frac{\left(2bc+ab+ac\right)^{2}}{b^{2}+4bc+c^{2}}+\frac{\left(2ca+bc+ba\right)^{2}}{c^{2}+4ca+a^{2}}\ge \frac{8(ab+bc+ca)}{3}.}$$
0 replies
KhuongTrang
32 minutes ago
0 replies
J
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circle geometry showing perpendicularity
Kyj9981   4
N 44 minutes ago by cj13609517288
Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A line through $B$ intersects $\omega_1$ and $\omega_2$ at points $C$ and $D$, respectively. Line $AD$ intersects $\omega_1$ at point $E \neq A$, and line $AC$ intersects $\omega_2$ at point $F \neq A$. If $O$ is the circumcenter of $\triangle AEF$, prove that $OB \perp CD$.
4 replies
1 viewing
Kyj9981
Mar 18, 2025
cj13609517288
44 minutes ago
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Prove excircle is tangent to circumcircle
sarjinius   8
N an hour ago by Lyzstudent
Source: Philippine Mathematical Olympiad 2025 P4
Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on side $BC$. Points $X$ and $Y$ are chosen on lines $BI$ and $CI$ respectively such that $DXIY$ is a parallelogram. Points $E$ and $F$ are chosen on side $BC$ such that $AX$ and $AY$ are the angle bisectors of angles $\angle BAE$ and $\angle CAF$ respectively. Let $\omega$ be the circle tangent to segment $EF$, the extension of $AE$ past $E$, and the extension of $AF$ past $F$. Prove that $\omega$ is tangent to the circumcircle of triangle $ABC$.
8 replies
sarjinius
Mar 9, 2025
Lyzstudent
an hour ago
J
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URGENT JMO problem 1 Misgrade protest (Cost MOP)
bjump   73
N an hour ago by NoSignOfTheta
I was docked 4 points on jmo 1 and it cost me mop. I got 370 777 and 770 777 got into mop.
This google drive link contains my submission to USAJMO day 1
Day 1 Scans
My solution works except for 2 typos. I wrote bijective instead of non bijective at the end, and i wrote min intead of more specifically minimum over Z. After discussion with vsamc, and megarnie they agreed I should have gotten a 7 on this problem because i demostrated that I knew how to solve it. Is it possible to protest my score, and get into MOP.

Help would be greatly appreciated :surrender:
73 replies
+2 w
bjump
Yesterday at 4:40 PM
NoSignOfTheta
an hour ago
J
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Subset coloring
v_Enhance   74
N an hour ago by de-Kirschbaum
Source: USAMO 2015 Problem 3
Let $S = \left\{ 1,2,\dots,n \right\}$, where $n \ge 1$. Each of the $2^n$ subsets of $S$ is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set $T \subseteq S$, we then write $f(T)$ for the number of subsets of $T$ that are blue.

Determine the number of colorings that satisfy the following condition: for any subsets $T_1$ and $T_2$ of $S$, \[ f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2). \]
74 replies
1 viewing
v_Enhance
Apr 28, 2015
de-Kirschbaum
an hour ago
J
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IMO Shortlist 2014 N6
hajimbrak   28
N an hour ago by MajesticCheese
Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$. On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$.

Proposed by Serbia
28 replies
hajimbrak
Jul 11, 2015
MajesticCheese
an hour ago
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3 knightlike moves is enough
sarjinius   3
N an hour ago by JollyEggsBanana
Source: Philippine Mathematical Olympiad 2025 P6
An ant is on the Cartesian plane. In a single move, the ant selects a positive integer $k$, then either travels [list]
[*] $k$ units vertically (up or down) and $2k$ units horizontally (left or right); or
[*] $k$ units horizontally (left or right) and $2k$ units vertically (up or down).
[/list]
Thus, for any $k$, the ant can choose to go to one of eight possible points.
Prove that, for any integers $a$ and $b$, the ant can travel from $(0, 0)$ to $(a, b)$ using at most $3$ moves.
3 replies
sarjinius
Mar 9, 2025
JollyEggsBanana
an hour ago
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USA(J)MO Statistics Out
BS2012   28
N 2 hours ago by KnowingAnt
Source: MAA edvistas page
https://maa.edvistas.com/eduview/report.aspx?view=1561&mode=6
who were the 2 usamo perfects
28 replies
BS2012
Yesterday at 10:07 PM
KnowingAnt
2 hours ago
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USAPhO Qualifiers
HungryCalculator   2
N Apr 14, 2025 by Narwhal234
Does anyone know why the 2025 USAPhO Qualifiers aren’t on the 2025 US Physics Team page?
2 replies
HungryCalculator
Apr 14, 2025
Narwhal234
Apr 14, 2025
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USAPhO Qualifiers
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Reveal topic tags
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HungryCalculator
539 posts
HungryCalculator
#1 h  Apr 14, 2025, 11:57 AM
Y by
Does anyone know why the 2025 USAPhO Qualifiers aren’t on the 2025 US Physics Team page?
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darkgrimoire
5 posts
darkgrimoire
#2 h  Apr 14, 2025, 1:30 PM
Y by
Not
yet released
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Narwhal234
31 posts
Narwhal234
#3 h  Apr 14, 2025, 9:35 PM
Y by
Takes a bit of time dw
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