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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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two circumcenters and one orthocenter, vertices of parallelogram
parmenides51   4
N 3 minutes ago by AylyGayypow009
Source: Greece JBMO TST 2015 p2
Let $ABC$ be an acute triangle inscribed in a circle of center $O$. If the altitudes $BD,CE$ intersect at $H$ and the circumcenter of $\triangle BHC$ is $O_1$, prove that $AHO_1O$ is a parallelogram.
4 replies
parmenides51
Apr 29, 2019
AylyGayypow009
3 minutes ago
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m^4+3^m is a perfect square number
Havu   5
N 6 minutes ago by MR.1
Find a positive integer m such that $m^4+3^m$ is a perfect square number.
5 replies
Havu
an hour ago
MR.1
6 minutes ago
J
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angle chasing with 2 midpoints, equal angles given and wanted
parmenides51   4
N 32 minutes ago by Blackbeam999
Source: Ukrainian Geometry Olympiad 2017, IX p1, X p1, XI p1
In the triangle $ABC$, ${{A}_{1}}$ and ${{C}_{1}} $ are the midpoints of sides $BC $ and $AB$ respectively. Point $P$ lies inside the triangle. Let $\angle BP {{C}_{1}} = \angle PCA$. Prove that $\angle BP {{A}_{1}} = \angle PAC $.
4 replies
parmenides51
Dec 11, 2018
Blackbeam999
32 minutes ago
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Determine all the 'good' numbers
April   4
N 34 minutes ago by DottedCaculator
Source: CGMO 2004 P1
We say a positive integer $ n$ is good if there exists a permutation $ a_1, a_2, \ldots, a_n$ of $ 1, 2, \ldots, n$ such that $ k + a_k$ is perfect square for all $ 1\le k\le n$. Determine all the good numbers in the set $ \{11, 13, 15, 17, 19\}$.
4 replies
April
Dec 27, 2008
DottedCaculator
34 minutes ago
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Classical factorial number theory
Orestis_Lignos   21
N 41 minutes ago by MIC38
Source: JBMO 2023 Problem 1
Find all pairs $(a,b)$ of positive integers such that $a!+b$ and $b!+a$ are both powers of $5$.

Nikola Velov, North Macedonia
21 replies
Orestis_Lignos
Jun 26, 2023
MIC38
41 minutes ago
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Functional equation
Nima Ahmadi Pour   100
N an hour ago by jasperE3
Source: ISl 2005, A2, Iran prepration exam
We denote by $\mathbb{R}^+$ the set of all positive real numbers.

Find all functions $f: \mathbb R^ + \rightarrow\mathbb R^ +$ which have the property:
\[f(x)f(y)=2f(x+yf(x))\]
for all positive real numbers $x$ and $y$.

Proposed by Nikolai Nikolov, Bulgaria
100 replies
Nima Ahmadi Pour
Apr 24, 2006
jasperE3
an hour ago
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how can I solve this FE
Jackson0423   6
N an hour ago by jasperE3

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function that satisfies the following equation for all real numbers \( x \):
\[ f(x^2 + x + 3) + 2f(x^2 - 3x + 5) = 6x^2 - 10x + 17. \]Find the value of \( f(100) \).
6 replies
Jackson0423
2 hours ago
jasperE3
an hour ago
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a does not divide b^n - n for all positive integers n?
N.T.TUAN   28
N an hour ago by alexanderhamilton124
Source: USA Team Selection Test 2007
Determine whether or not there exist positive integers $ a$ and $ b$ such that $ a$ does not divide $ b^n - n$ for all positive integers $ n$.
28 replies
N.T.TUAN
Dec 8, 2007
alexanderhamilton124
an hour ago
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Urgent. Need them quick
sealight2107   1
N an hour ago by arqady
With $a,b,c>1$ and $a+b+c=2abc$. Prove that:
$\sqrt[3]{ab-1}+\sqrt[3]{bc-1}+\sqrt[3]{ca-1} \le \sqrt[3]{(a+b+c)^2}$
1 reply
sealight2107
an hour ago
arqady
an hour ago
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Divisibility..
Sadigly   3
N an hour ago by Jackson0423
Source: Azerbaijan Junior MO 2025 P2
Find all $4$ consecutive even numbers, such that the square of their product is divisible by the sum of their squares.
3 replies
Sadigly
Today at 7:37 AM
Jackson0423
an hour ago
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Divisibilty...
Sadigly   9
N 2 hours ago by Jackson0423
Source: My (fake) translation error
Find all $4$ consecutive even numbers, such that the square of their product divides the sum of their squares.
9 replies
Sadigly
2 hours ago
Jackson0423
2 hours ago
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Inspired by Kosovo 2010
sqing   1
N 2 hours ago by ytChen
Source: Own
Let $ a,b>0 , a+b\leq k $. Prove that
$$\left(1+\frac{1}{a(b+1)}\right)\left(1+\frac{1}{b(a+1)}\right)\geq\left(1+\frac{4}{k(k+2)}\right)^2$$$$\left(1+\frac {a}{b(a+1)}\right)\left(1+\frac {b}{a(b+1)}\right) \geq\left(1+\frac{2}{k+2}\right)^2$$Let $ a,b>0 , a+b\leq 2 $. Prove that
$$\left(1+\frac{1}{a(b+1)}\right)\left(1+\frac{1}{b(a+1)}\right)\geq \frac{9}{4} $$$$\left(1+\frac {a}{b(a+1)}\right)\left(1+\frac {b}{a(b+1)}\right) \geq \frac{9}{4} $$
1 reply
sqing
Today at 3:56 AM
ytChen
2 hours ago
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Number Theory
VicKmath7   5
N 2 hours ago by Adywastaken
Source: Archimedes Junior 2014
Let $p$ prime and $m$ a positive integer. Determine all pairs $( p,m)$ satisfying the equation: $ p(p+m)+p=(m+1)^3$
5 replies
VicKmath7
Mar 17, 2020
Adywastaken
2 hours ago
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x+y in B iff x,y in A
fattypiggy123   5
N 2 hours ago by Math2030
Source: China Mathematical Olympiad 2015 Q3
Let $n \geq 5$ be a positive integer and let $A$ and $B$ be sets of integers satisfying the following conditions:

i) $|A| = n$, $|B| = m$ and $A$ is a subset of $B$
ii) For any distinct $x,y \in B$, $x+y \in B$ iff $x,y \in A$

Determine the minimum value of $m$.
5 replies
fattypiggy123
Dec 20, 2014
Math2030
2 hours ago
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The smallest of sum of elements
hlminh   1
N Apr 23, 2025 by nguyenhuybao_06
Let $S=\{1,2,...,2014\}$ and $X\subset S$ such that for all $a,b\in X,$ if $a+b\leq 2014$ then $a+b\in X.$ Find the smallest of sum of all elements of $X.$
1 reply
hlminh
Apr 23, 2025
nguyenhuybao_06
Apr 23, 2025
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The smallest of sum of elements
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combinatorics
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hlminh
240 posts
hlminh
#1 h  Apr 23, 2025, 6:33 PM
Y by
Let $S=\{1,2,...,2014\}$ and $X\subset S$ such that for all $a,b\in X,$ if $a+b\leq 2014$ then $a+b\in X.$ Find the smallest of sum of all elements of $X.$
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nguyenhuybao_06
39 posts
nguyenhuybao_06
#2 h  Apr 23, 2025, 6:44 PM
Y by
This problem is tricky lol.
If $a+b\in X, a+b\le 2024,$ then $a+b \in X.$ Note that since $X$ is a finite set, there exists $x,y\in X$ such that $x+y\ge 2025.$ So that the smallest of sum of all elements of $X$ is $2025.$
For the construction, $X=\lbrace1007,1008\rbrace.$
This post has been edited 1 time. Last edited by nguyenhuybao_06, Apr 23, 2025, 6:44 PM
Reason: typo
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