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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
1 viewing
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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Classic graph theory lemma?
eulerleonhardfan   1
N 6 minutes 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
11 minutes ago
eulerleonhardfan
6 minutes ago
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circle geometry showing perpendicularity
Kyj9981   3
N 16 minutes 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
Kyj9981
Mar 18, 2025
JollyEggsBanana
16 minutes ago
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Min Number of Subsets of Strictly Increasing
taptya17   5
N 18 minutes 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
18 minutes ago
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Nice inequality
sqing   3
N 24 minutes 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
1 viewing
sqing
Apr 24, 2019
Oksutok
24 minutes ago
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Inspired by 2024 Fall LMT Guts
sqing   2
N 26 minutes 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
an hour ago
Jackson0423
26 minutes ago
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Dividing Pairs
Jackson0423   2
N 42 minutes 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
42 minutes ago
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Ellipse and Vectors
scls140511   1
N an hour ago by Tigolf
Source: 2024 China Round 1 (Gao Lian)
7 Let $F_1$ and $F_2$ be the two foci of ellipse $\omega$. $P$ is a point on $\omega$. Let $O$ be the center of the excircle of $\triangle PF_1F_2$. When $\vec{PO} \cdot \vec{F_1F_2} = 2\vec{PF_1} \cdot \vec{PF_2}$, find the minimum eccentricity of $\omega$.
1 reply
1 viewing
scls140511
Sep 8, 2024
Tigolf
an hour ago
J
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Maximum number of nice subsets
FireBreathers   1
N an hour ago by FireBreathers
Given a set $M$ of natural numbers with $n$ elements with $n$ odd number. A nonempty subset $S$ of $M$ is called $nice$ if the product of the elements of $S$ divisible by the sum of the elements of $M$, but not by its square. It is known that the set $M$ itself is good. Determine the maximum number of $nice$ subsets (including $M$ itself).
1 reply
FireBreathers
Yesterday at 10:27 PM
FireBreathers
an hour ago
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Existence of reals satisfying cyclic relation
DVDthe1st   10
N an hour ago by sttsmet
Source: 2018 China TST Day 1 Q1
Let $p,q$ be positive reals with sum 1. Show that for any $n$-tuple of reals $(y_1,y_2,...,y_n)$, there exists an $n$-tuple of reals $(x_1,x_2,...,x_n)$ satisfying $$p\cdot \max\{x_i,x_{i+1}\} + q\cdot \min\{x_i,x_{i+1}\} = y_i$$for all $i=1,2,...,2017$, where $x_{2018}=x_1$.
10 replies
DVDthe1st
Jan 2, 2018
sttsmet
an hour ago
J
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How many non-attacking pawns can be placed on a $n \times n$ chessboard?
DylanN   2
N an hour ago by zRevenant
Source: 2019 Pan-African Shortlist - C1
A pawn is a chess piece which attacks the two squares diagonally in front if it. What is the maximum number of pawns which can be placed on an $n \times n$ chessboard such that no two pawns attack each other?
2 replies
DylanN
Jan 18, 2021
zRevenant
an hour ago
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Solve All 6 IMO 2024 Problems (42/42), New Framework Looking for Feedback
Blackhole.LightKing   0
an hour ago
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


0 replies
1 viewing
Blackhole.LightKing
an hour ago
0 replies
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Russian NT with a Ceiling
naman12   45
N 2 hours ago by InterLoop
Source: 2019 ISL N8
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)

Russia
45 replies
naman12
Sep 22, 2020
InterLoop
2 hours ago
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Excircle Tangency Points Concyclic with A
tastymath75025   35
N 2 hours ago by bin_sherlo
Source: USA Winter TST for IMO 2019, Problem 6, by Ankan Bhattacharya
Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on line $BC$ satisfying $\angle AID=90^{\circ}$. Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to $\overline{BC}$ at $A_1$. Define points $B_1$ on $\overline{CA}$ and $C_1$ on $\overline{AB}$ analogously, using the excircles opposite $B$ and $C$, respectively.

Prove that if quadrilateral $AB_1A_1C_1$ is cyclic, then $\overline{AD}$ is tangent to the circumcircle of $\triangle DB_1C_1$.

Ankan Bhattacharya
35 replies
tastymath75025
Jan 21, 2019
bin_sherlo
2 hours ago
J
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Inspired by SXTX (4)2025 Q712
sqing   0
2 hours ago
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$$ abc(a+b+c) \leq \frac{9}{8} $$
0 replies
sqing
2 hours ago
0 replies
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J
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Convex pentagon - A, B, C, D are concyclic
lssl   1
N Sep 18, 2010 by jgnr
In the convex pentagon $ABCDE$ we know that $AB=DE, BC=EA$ but $AB \neq EA$.
$B,C,D,E$ are concyclic .
Prove that $A,B,C,D$ are concyclic if and only if $AC=AD.$
1 reply
lssl
Sep 18, 2010
jgnr
Sep 18, 2010
J
Convex pentagon - A, B, C, D are concyclic
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Reveal topic tags
geometry
trapezoid
circumcircle
geometry proposed
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lssl
240 posts
lssl
#1 h  Sep 18, 2010, 10:24 AM • 1 Y
Y by Adventure10
In the convex pentagon $ABCDE$ we know that $AB=DE, BC=EA$ but $AB \neq EA$.
$B,C,D,E$ are concyclic .
Prove that $A,B,C,D$ are concyclic if and only if $AC=AD.$
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jgnr
1343 posts
jgnr
#2 h  Sep 18, 2010, 2:39 PM • 2 Y
Y by Adventure10, Mango247
Proof of necessicity: Suppose $A,B,C,D$ are concyclic. $AB=DE$ and $BC=EA$ implies that $ABDE$ and $ABCE$ are trapezoid. So $AC=BE=AD$.

Proof of sufficiency: Suppose $AC=AD$. Triangles $ABC$ and $DEA$ are congruent. Let $F$ be the point such that $CDEF$ is an isosceles trapezoid ($EF\parallel CD$). Hence $B,C,D,E,F$ are concyclic. We also have $\angle ACF=\angle DAE=\angle BAC$ and $CF=DE=AB$, hence $ABFC$ is an isosceles trapezoid, which implies that $A$ also lies on the circumcircle of $BCDEF$. Thus $A,B,C,D$ are concyclic.
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