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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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[*]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
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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Prove excircle is tangent to circumcircle
sarjinius   8
N 5 minutes 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
5 minutes ago
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Solve All 6 IMO 2024 Problems (42/42), New Framework Looking for Feedback
Blackhole.LightKing   1
N 7 minutes ago by GreekIdiot
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


1 reply
Blackhole.LightKing
2 hours ago
GreekIdiot
7 minutes ago
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Number Theory
AnhQuang_67   1
N 10 minutes ago by GreekIdiot
Source: HSGSO 2024
Let $p$ be an even prime number and a sequence $\{a_n\}_{n=1}^{+\infty}$ satisfy $$a_1=1, a_2=2$$and $$a_{n+2}=2\cdot a_{n+1}+3\cdot a_n, \forall n \geqslant 1$$Prove that always exists positive integer $k$ satisfying for all positive integers $n$, then $a_n \ne k \mod{p}$.

P/s: $\ne$ is "not congruence"
1 reply
AnhQuang_67
an hour ago
GreekIdiot
10 minutes ago
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Irrational equation
giangtruong13   1
N 15 minutes ago by Tuvshuu
Solve the equation : $$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$
1 reply
giangtruong13
an hour ago
Tuvshuu
15 minutes ago
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IMO Shortlist 2014 N6
hajimbrak   28
N 30 minutes 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
30 minutes ago
J
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3 knightlike moves is enough
sarjinius   3
N 35 minutes 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
35 minutes ago
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Why is the old one deleted?
EeEeRUT   15
N 38 minutes ago by Tuvshuu
Source: EGMO 2025 P1
For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$for all $1 \leqslant i \leqslant m-1$

Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.

Proposed by Paulius Aleknavičius, Lithuania
15 replies
EeEeRUT
Apr 16, 2025
Tuvshuu
38 minutes ago
J
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My problem that I could not find(NT)
Nuran2010   0
an hour ago
Source: Own
While I was thinking on some other geometry problem, a NT problem came to my mind. Despite some tries(which were mostly order), I could not find a way to solve the problem. As I searched, this problem has never been posted before. Here is the problem.

Find all positive integers $a,b$ such that:
$a+b|2^{ab}+1$

Moreover, I wonder if there is a way to solve the question in this variant:

Find all positive integers $a,b,n$ such that:
$a+b|n^{ab}+1$
0 replies
Nuran2010
an hour ago
0 replies
J
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Classic graph theory lemma?
eulerleonhardfan   1
N an hour ago by eulerleonhardfan
$n \in \mathbb{N}$ is given, $A$, $B$ are graphs on the same set of $n$ nodes, having $a, b$ connected components respectively. Prove that $A \cup B$ has at least $a+b-n$ connected components.
1 reply
eulerleonhardfan
an hour ago
eulerleonhardfan
an hour ago
J
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circle geometry showing perpendicularity
Kyj9981   3
N an hour ago by JollyEggsBanana
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 $OB \perp CD$.
3 replies
+1 w
Kyj9981
Mar 18, 2025
JollyEggsBanana
an hour ago
J
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Min Number of Subsets of Strictly Increasing
taptya17   5
N an hour ago by kotmhn
Source: India EGMO TST 2025 Day 1 P1
Let $n$ be a positive integer. Initially the sequence $0,0,\cdots,0$ ($n$ times) is written on the board. In each round, Ananya choses an integer $t$ and a subset of the numbers written on the board and adds $t$ to all of them. What is the minimum number of rounds in which Ananya can make the sequence on the board strictly increasing?

Proposed by Shantanu Nene
5 replies
taptya17
Dec 13, 2024
kotmhn
an hour ago
J
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Nice inequality
sqing   3
N an hour ago by Oksutok
Source: WYX
Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be real numbers . Prove that : There exist positive integer $k\in \{1,2,\cdots,n\}$ such that $$\sum_{i=1}^{n}\{kx_i\}(1-\{kx_i\})<\frac{n-1}{6}.$$Where $\{x\}=x-\left \lfloor x \right \rfloor.$
3 replies
sqing
Apr 24, 2019
Oksutok
an hour ago
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Inspired by 2024 Fall LMT Guts
sqing   2
N an hour ago by Jackson0423
Source: Own
Let $x$, $y$, $z$ are pairwise distinct real numbers satisfying $x^2+y =y^2 +z = z^2+x. $ Prove that
$$(x+y)(y+z)(z+x)=-1$$Let $x$, $y$, $z$ are pairwise distinct real numbers satisfying $x^2+2y =y^2 +2z = z^2+2x. $ Prove that
$$(x+y)(y+z)(z+x)=-8$$
2 replies
sqing
2 hours ago
Jackson0423
an hour ago
J
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Dividing Pairs
Jackson0423   2
N 2 hours ago by Jackson0423
Source: Own
Let \( a \) and \( b \) be positive integers.
Suppose that \( a \) is a divisor of \( b^2 + 1 \) and \( b \) is a divisor of \( a^2 + 1 \).
Find all such pairs \( (a, b) \).
2 replies
Jackson0423
Apr 13, 2025
Jackson0423
2 hours ago
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J
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|AP|+ |BC| =\sqrt3 |AB| if <ABP = 20^o, <PBC =<PCB = 10^o, <PCA = 40^o
parmenides51   3
N Dec 17, 2021 by BarisKoyuncu
Source: Romanian Mathematical Magazine JP 353 https://artofproblemsolving.com/community/c2691230_
In $\vartriangle ABC$, $P \in Int (\vartriangle ABC)$, $\angle ABP = 20^o$, $\angle PBC =\angle PCB = 10^o$, $\angle PCA = 40^o$. Prove that $|AP|+ |BC| =\sqrt3 |AB|$.

by Mehmet Sahin - Turkey
3 replies
parmenides51
Dec 14, 2021
BarisKoyuncu
Dec 17, 2021
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|AP|+ |BC| =\sqrt3 |AB| if <ABP = 20^o, <PBC =<PCB = 10^o, <PCA = 40^o
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Reveal topic tags
angles
geometry
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Source: Romanian Mathematical Magazine JP 353 https://artofproblemsolving.com/community/c2691230_
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parmenides51
30630 posts
parmenides51
#1 h  Dec 14, 2021, 9:45 PM
Y by
In $\vartriangle ABC$, $P \in Int (\vartriangle ABC)$, $\angle ABP = 20^o$, $\angle PBC =\angle PCB = 10^o$, $\angle PCA = 40^o$. Prove that $|AP|+ |BC| =\sqrt3 |AB|$.

by Mehmet Sahin - Turkey
This post has been edited 1 time. Last edited by parmenides51, Dec 14, 2021, 9:45 PM
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sunken rock
4384 posts
sunken rock
#2 h  Dec 17, 2021, 7:10 AM
Y by
Nobody yet? Come on!
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BarisKoyuncu
577 posts
BarisKoyuncu
#3 h  Dec 17, 2021, 7:19 AM
Y by
Trigonometric solution
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BarisKoyuncu
577 posts
BarisKoyuncu
#4 h  Dec 17, 2021, 7:54 AM
Y by
Non-trigonometric solution
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