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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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Functional equation of nonzero reals
proglote   7
N 6 minutes ago by jasperE3
Source: Brazil MO 2013, problem #3
Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.
7 replies
proglote
Oct 24, 2013
jasperE3
6 minutes ago
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Interesting inequalities
sqing   0
22 minutes ago
Source: Own
Let $ a,b\geq 0 ,a^2-ab+b^2+a+b=3 $. Prove that
$$ \frac{39+\sqrt{13}}{78}\geq \frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \geq \frac{1}{2}$$Let $ a,b\geq 0 ,a^2+ab+b^2+a+b=3 $. Prove that
$$ \frac{19+\sqrt{10}}{39}\geq \frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \geq \frac{39+\sqrt{13}}{78}$$Let $ a,b\geq 0 ,a^2+ab+b^2+a+b=5 $. Prove that
$$ \frac{3}{5}> \frac{1}{a^2+3}+ \frac{1}{b^2+ 3} \geq \frac{185+3\sqrt{21}}{402}$$
0 replies
sqing
22 minutes ago
0 replies
J
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4-var inequality
sqing   2
N an hour ago by sqing
Source: SXTB (4)2025 Q2837
Let $ a,b,c,d> 0 $. Prove that
$$ \frac{1}{(3a+1)^4}+ \frac{1}{(3b+1)^4}+\frac{1}{(3c+1)^4}+\frac{1}{(3d+1)^4} \geq \frac{1}{16(3abcd+1)}$$
2 replies
sqing
Yesterday at 2:59 PM
sqing
an hour ago
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Inspired by Bet667
sqing   2
N an hour ago by sqing
Source: Own
Let $ a,b $ be a real numbers such that $a^2+kab+b^2\ge a^3+b^3.$Prove that$$a+b\leq k+2$$Where $ k\geq 0. $
2 replies
sqing
Yesterday at 2:46 PM
sqing
an hour ago
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find positive n so that exists prime p with p^n-(p-1)^n$ a power of 3
parmenides51   14
N an hour ago by MathLuis
Source: JBMO Shortlist 2017 NT5
Find all positive integers $n$ such that there exists a prime number $p$, such that $p^n-(p-1)^n$ is a power of $3$.

Note. A power of $3$ is a number of the form $3^a$ where $a$ is a positive integer.
14 replies
parmenides51
Jul 25, 2018
MathLuis
an hour ago
J
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GCD of terms in a sequence
BBNoDollar   1
N an hour ago by mashumaro
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
1 reply
BBNoDollar
5 hours ago
mashumaro
an hour ago
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Diodes and usamons
v_Enhance   46
N 2 hours ago by HamstPan38825
Source: USA December TST for the 56th IMO, by Linus Hamilton
A physicist encounters $2015$ atoms called usamons. Each usamon either has one electron or zero electrons, and the physicist can't tell the difference. The physicist's only tool is a diode. The physicist may connect the diode from any usamon $A$ to any other usamon $B$. (This connection is directed.) When she does so, if usamon $A$ has an electron and usamon $B$ does not, then the electron jumps from $A$ to $B$. In any other case, nothing happens. In addition, the physicist cannot tell whether an electron jumps during any given step. The physicist's goal is to isolate two usamons that she is sure are currently in the same state. Is there any series of diode usage that makes this possible?

Proposed by Linus Hamilton
46 replies
v_Enhance
Dec 17, 2014
HamstPan38825
2 hours ago
J
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China South East Mathematical Olympiad 2013 problem 2
s372102   3
N 2 hours ago by AGCN
$\triangle ABC$, $AB>AC$. the incircle $I$ of $\triangle ABC$ meet $BC$ at point $D$, $AD$ meet $I$ again at $E$. $EP$ is a tangent of $I$, and $EP$ meet the extension line of $BC$ at $P$. $CF\parallel PE$, $CF\cap AD=F$. the line $BF$ meet $I$ at $M,N$, point $M$ is on the line segment $BF$, the line segment $PM$ meet $I$ again at $Q$. Show that $\angle ENP=\angle ENQ$
3 replies
s372102
Aug 10, 2013
AGCN
2 hours ago
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f(x+y) = max(f(x), y) + min(f(y), x)
Zhero   50
N 2 hours ago by lpieleanu
Source: ELMO Shortlist 2010, A3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$.

George Xing.
50 replies
Zhero
Jul 5, 2012
lpieleanu
2 hours ago
J
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Similar to iran 1996
GreekIdiot   1
N 2 hours ago by Lufin
Let $f: \mathbb R \to \mathbb R$ be a function such that $f(f(x)+y)=f(f(x)-y)+4f(x)y \: \forall x,y \: \in \: \mathbb R$. Find all such $f$.
1 reply
GreekIdiot
Apr 26, 2025
Lufin
2 hours ago
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Prove that 4p-3 is a square - Iran NMO 2005 - Problem1
sororak   24
N 2 hours ago by EmersonSoriano
Let $n,p>1$ be positive integers and $p$ be prime. We know that $n|p-1$ and $p|n^3-1$. Prove that $4p-3$ is a perfect square.
24 replies
sororak
Sep 21, 2010
EmersonSoriano
2 hours ago
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IMO ShortList 1999, combinatorics problem 4
orl   27
N 2 hours ago by cursed_tangent1434
Source: IMO ShortList 1999, combinatorics problem 4
Let $A$ be a set of $N$ residues $\pmod{N^{2}}$. Prove that there exists a set $B$ of of $N$ residues $\pmod{N^{2}}$ such that $A + B = \{a+b|a \in A, b \in B\}$ contains at least half of all the residues $\pmod{N^{2}}$.
27 replies
orl
Nov 14, 2004
cursed_tangent1434
2 hours ago
J
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Geometry
Lukariman   2
N 2 hours ago by Lukariman
Given circle (O) and point P outside (O). From P draw tangents PA and PB to (O) with contact points A, B. On the opposite ray of ray BP, take point M. The circle circumscribing triangle APM intersects (O) at the second point D. Let H be the projection of B on AM. Prove that <HDM = 2∠AMP.
2 replies
Lukariman
Yesterday at 12:43 PM
Lukariman
2 hours ago
J
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IMO 2010 Problem 1
canada   119
N 2 hours ago by lpieleanu
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[ f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$

Proposed by Pierre Bornsztein, France
119 replies
canada
Jul 7, 2010
lpieleanu
2 hours ago
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J
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Geometry inmo problem
Enes040612   2
N Apr 25, 2025 by Red_gear5246
Source: INMO 2008
Can anyone help with this problem? the book also starts the solution by saying IA1=IB1=IC1=2r which I'm not understanding why. The problem is at the attachments.
2 replies
Enes040612
Feb 13, 2025
Red_gear5246
Apr 25, 2025
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Geometry inmo problem
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Reveal topic tags
geometry
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G H BBookmark kLocked kLocked NReply
Source: INMO 2008
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Enes040612
4 posts
Enes040612
#1 h  Feb 13, 2025, 8:10 AM
Y by
Can anyone help with this problem? the book also starts the solution by saying IA1=IB1=IC1=2r which I'm not understanding why. The problem is at the attachments.
Attachments:
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CHESSR1DER
58 posts
CHESSR1DER
#2 h  Feb 13, 2025, 9:20 AM
Y by
Let $IA1 \cap BC = H$. Then since $IH = r$ and $IH = A1H$, then $IA1= 2r$. Other are similar.
This post has been edited 2 times. Last edited by CHESSR1DER, Feb 13, 2025, 9:21 AM
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Red_gear5246
2 posts
Red_gear5246
#3 h  Apr 25, 2025, 1:29 PM
Y by
@Enes040612 which book are u solving?
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