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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 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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help me~~
Imshyso   0
24 minutes ago
Given triangle ABC inscribed in (O) with orthocenter H. Let K be the midpoint of AH. Take E,F on AC, AB so that BKE=CKF=90. Prove that E,O,F are collinear.
0 replies
+1 w
Imshyso
24 minutes ago
0 replies
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Basic ideas in junior diophantine equations
Maths_VC   5
N 38 minutes ago by Royal_mhyasd
Source: Serbia JBMO TST 2025, Problem 3
Determine all positive integers $a, b$ and $c$ such that
$2$ $\cdot$ $10^a + 5^b = 2025^c$
5 replies
Maths_VC
May 27, 2025
Royal_mhyasd
38 minutes ago
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Find values for a, b ,c
Ferum_2710   2
N 39 minutes ago by Jupiterballs
Source: Romania JBMO tst 2023 day2 p4
Let $M \geq 1$ be a real number. Determine all natural numbers $n$ for which there exist distinct natural numbers $a$, $b$, $c > M$, such that
$n = (a,b) \cdot (b,c) + (b,c) \cdot (c,a) + (c,a) \cdot (a,b)$
(where $(x,y)$ denotes the greatest common divisor of natural numbers $x$ and $y$).
2 replies
Ferum_2710
Apr 30, 2023
Jupiterballs
39 minutes ago
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Center lies on circumcircle of other
Philomath_314   41
N an hour ago by Adywastaken
Source: INMO P1
In triangle $ABC$ with $CA=CB$, point $E$ lies on the circumcircle of $ABC$ such that $\angle ECB=90^{\circ}$. The line through $E$ parallel to $CB$ intersects $CA$ in $F$ and $AB$ in $G$. Prove that the center of the circumcircle of triangle $EGB$ lies on the circumcircle of triangle $ECF$.

Proposed by Prithwijit De
41 replies
Philomath_314
Jan 21, 2024
Adywastaken
an hour ago
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find question
mathematical-forest   7
N an hour ago by whwlqkd
Are there any contest questions that seem simple but are actually difficult? :-D
7 replies
mathematical-forest
Thursday at 10:19 AM
whwlqkd
an hour ago
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Inspired by a cool result
DoThinh2001   1
N an hour ago by arqady
Source: Old?
Let three real numbers $a,b,c\geq 0$, no two of which are $0$. Prove that:
$$\sqrt{\frac{a^2+bc}{b^2+c^2}}+\sqrt{\frac{b^2+ca}{c^2+a^2}}+\sqrt{\frac{c^2+ab}{a^2+b^2}}\geq 2+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}.$$
Inspiration
1 reply
+1 w
DoThinh2001
Today at 12:08 AM
arqady
an hour ago
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Crossing ٍٍChords
matinyousefi   1
N 2 hours ago by Trenod
Source: Iranian Combinatorics Olympiad 2020 P3
$1399$ points and some chords between them is given.
$a)$ In every step we can take two chords $RS,PQ$ with a common point other than $P,Q,R,S$ and erase exactly one of $RS,PQ$ and draw $PS,PR,QS,QR$ let $s$ be the minimum of chords after some steps. Find the maximum of $s$ over all initial positions.
$b)$ In every step we can take two chords $RS,PQ$ with a common point other than $P,Q,R,S$ and erase both of $RS,PQ$ and draw $PS,PR,QS,QR$ let $s$ be the minimum of chords after some steps. Find the maximum of $s$ over all initial positions.

Proposed by Afrouz Jabalameli, Abolfazl Asadi
1 reply
matinyousefi
Apr 24, 2020
Trenod
2 hours ago
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Nice NT with powers of two
oVlad   7
N 2 hours ago by SimplisticFormulas
Source: Romania TST 2024 Day 1 P3
Let $n{}$ be a positive integer and let $a{}$ and $b{}$ be positive integers congruent to 1 modulo 4. Prove that there exists a positive integer $k{}$ such that at least one of the numbers $a^k-b$ and $b^k-a$ is divisible by $2^n.$

Cătălin Liviu Gherghe
7 replies
oVlad
Jul 31, 2024
SimplisticFormulas
2 hours ago
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D,E,F are collinear.
TUAN2k8   2
N 2 hours ago by TUAN2k8
Source: Own
Help me with this:
2 replies
TUAN2k8
May 28, 2025
TUAN2k8
2 hours ago
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Combinatorial identity
MehdiGolafshan   4
N 2 hours ago by watery
Let $n$ is a positive integer. Prove that
$$\sum_{k=0}^{n-1}\frac{1}{k+1}\binom{n-1}{k} = \frac{2^n-1}{n}.$$
4 replies
MehdiGolafshan
Jan 16, 2023
watery
2 hours ago
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Handouts/Resources on Limits.
Saucepan_man02   0
5 hours ago
Could anyone kindly share some resources/handouts on limits?
0 replies
Saucepan_man02
5 hours ago
0 replies
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IMC 1994 D2 P1
j___d   13
N Yesterday at 11:20 PM by krigger
Let $f\in C^1[a,b]$, $f(a)=0$ and suppose that $\lambda\in\mathbb R$, $\lambda >0$, is such that
$$|f'(x)|\leq \lambda |f(x)|$$for all $x\in [a,b]$. Is it true that $f(x)=0$ for all $x\in [a,b]$?
13 replies
j___d
Mar 6, 2017
krigger
Yesterday at 11:20 PM
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D1039 : A strange and general result on series
Dattier   0
Yesterday at 10:33 PM
Source: les dattes à Dattier
Let $f \in C([0,1];[0,1])$ bijective, $f(0)=0$ and $(a_k) \in [0,1]^\mathbb N$ with $ \sum \limits_{k=0}^{+\infty} a_k$ converge.

Is it true that $\sum \limits_{k=0}^{+\infty} f(a_k)\times f^{-1}(a_k)$ converge?
0 replies
Dattier
Yesterday at 10:33 PM
0 replies
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Aproximate ln(2) using perfect numbers
YLG_123   5
N Yesterday at 8:55 PM by ei_killua_
Source: Brazilian Mathematical Olympiad 2024, Level U, Problem 1
A positive integer \(n\) is called perfect if the sum of its positive divisors \(\sigma(n)\) is twice \(n\), that is, \(\sigma(n) = 2n\). For example, \(6\) is a perfect number since the sum of its positive divisors is \(1 + 2 + 3 + 6 = 12\), which is twice \(6\). Prove that if \(n\) is a positive perfect integer, then:
\[ \sum_{p|n} \frac{1}{p + 1} < \ln 2 < \sum_{p|n} \frac{1}{p - 1} \]where the sums are taken over all prime divisors \(p\) of \(n\).
5 replies
YLG_123
Oct 12, 2024
ei_killua_
Yesterday at 8:55 PM
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Triple Sum
P162008   1
N Apr 28, 2025 by soryn
Source: From a friend
Evaluate $\Omega = \lim_{n \to \infty} \frac{1}{n^5} \sum_{i=1}^{n} \sum_{k=1}^{2n} \sum_{k=1}^{3n} \frac{i(jk + 6n^2 - 3jn - 2kn) - jkn + n^2(3j + 2k) - 6n^3}{\sqrt{i^2 + j^2 + k^2 - i(a - 2n) + j(b - 4n) + k(c - 6n) + c + 14n^2}}$
1 reply
P162008
Apr 26, 2025
soryn
Apr 28, 2025
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Triple Sum
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Reveal topic tags
calculus
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Source: From a friend
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P162008
232 posts
P162008
#1 h  Apr 26, 2025, 9:37 AM
Y by
Evaluate $\Omega = \lim_{n \to \infty} \frac{1}{n^5} \sum_{i=1}^{n} \sum_{k=1}^{2n} \sum_{k=1}^{3n} \frac{i(jk + 6n^2 - 3jn - 2kn) - jkn + n^2(3j + 2k) - 6n^3}{\sqrt{i^2 + j^2 + k^2 - i(a - 2n) + j(b - 4n) + k(c - 6n) + c + 14n^2}}$
This post has been edited 1 time. Last edited by P162008, Apr 26, 2025, 9:38 AM
Reason: Typo
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soryn
5348 posts
soryn
#2 h  Apr 28, 2025, 8:11 PM
Y by
Triple integral?
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