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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
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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binomial sum ratio
thewayofthe_dragon   3
N an hour ago by P162008
Source: YT
Someone please evaluate this ratio inside the log for any given n(I feel the sum doesn't have any nice closed form).
3 replies
thewayofthe_dragon
Jun 16, 2024
P162008
an hour ago
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IMO Shortlist 2013, Combinatorics #3
lyukhson   31
N an hour ago by Maximilian113
Source: IMO Shortlist 2013, Combinatorics #3
A crazy physicist discovered a new kind of particle wich he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a time.
(i) If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it.
(ii) At any moment, he may double the whole family of imons in the lab by creating a copy $I'$ of each imon $I$. During this procedure, the two copies $I'$ and $J'$ become entangled if and only if the original imons $I$ and $J$ are entangled, and each copy $I'$ becomes entangled with its original imon $I$; no other entanglements occur or disappear at this moment.

Prove that the physicist may apply a sequence of such operations resulting in a family of imons, no two of which are entangled.
31 replies
lyukhson
Jul 9, 2014
Maximilian113
an hour ago
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A wizard kidnaps 101 people
Leicich   14
N an hour ago by MathCosine
Source: Argentina TST 2011, Problem 2
A wizard kidnaps $31$ members from party $A$, $28$ members from party $B$, $23$ members from party $C$, and $19$ members from party $D$, keeping them isolated in individual rooms in his castle, where he forces them to work.
Every day, after work, the kidnapped people can walk in the park and talk with each other. However, when three members of three different parties start talking with each other, the wizard reconverts them to the fourth party (there are no conversations with $4$ or more people involved).

a) Find out whether it is possible that, after some time, all of the kidnapped people belong to the same party. If the answer is yes, determine to which party they will belong.
b) Find all quartets of positive integers that add up to $101$ that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.
14 replies
1 viewing
Leicich
Aug 29, 2014
MathCosine
an hour ago
J
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PAMO Problem 4: Perpendicular lines
DylanN   11
N 2 hours ago by ATM_
Source: 2019 Pan-African Mathematics Olympiad, Problem 4
The tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$ meet at $D$. The circumcircle of $\triangle BCD$ meets sides $AC$ and $AB$ again at $E$ and $F$ respectively. Let $O$ be the circumcentre of $\triangle ABC$. Show that $AO$ is perpendicular to $EF$.
11 replies
DylanN
Apr 9, 2019
ATM_
2 hours ago
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Number Theory
fasttrust_12-mn   5
N 2 hours ago by GreekIdiot
Source: Pan African Mathematics Olympiad p6
Find all integers $n$ for which $n^7-41$ is the square of an integer
5 replies
fasttrust_12-mn
Aug 16, 2024
GreekIdiot
2 hours ago
J
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Maximum number of nice subsets
FireBreathers   0
2 hours ago
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).
0 replies
FireBreathers
2 hours ago
0 replies
J
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Floor double summation
CyclicISLscelesTrapezoid   52
N 2 hours ago by lpieleanu
Source: ISL 2021 A2
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]true?
52 replies
CyclicISLscelesTrapezoid
Jul 12, 2022
lpieleanu
2 hours ago
J
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Polynomial
Z_.   1
N 2 hours ago by rchokler
Let \( m \) be an integer greater than zero. Then, the value of the sum of the reciprocals of the cubes of the roots of the equation
\[ mx^4 + 8x^3 - 139x^2 - 18x + 9 = 0 \]is equal to:
1 reply
Z_.
3 hours ago
rchokler
2 hours ago
J
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Existence of perfect squares
egxa   2
N 2 hours ago by pavel kozlov
Source: All Russian 2025 10.3
Find all natural numbers \(n\) for which there exists an even natural number \(a\) such that the number
\[ (a - 1)(a^2 - 1)\cdots(a^n - 1) \]is a perfect square.
2 replies
egxa
Apr 18, 2025
pavel kozlov
2 hours ago
J
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IMO 2014 Problem 4
ipaper   169
N 4 hours ago by YaoAOPS
Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Let $M$ and $N$ be the points on $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$ and $Q$ is the midpoint of $AN$. Prove that the intersection of $BM$ and $CN$ is on the circumference of triangle $ABC$.

Proposed by Giorgi Arabidze, Georgia.
169 replies
ipaper
Jul 9, 2014
YaoAOPS
4 hours ago
J
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Inequalities
Scientist10   1
N 4 hours ago by Bergo1305
If $x, y, z \in \mathbb{R}$, then prove that the following inequality holds:
\[ \sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x \]
1 reply
Scientist10
Yesterday at 6:36 PM
Bergo1305
4 hours ago
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Tangents forms triangle with two times less area
NO_SQUARES   1
N 4 hours ago by Luis González
Source: Kvant 2025 no. 2 M2831
Let $DEF$ be triangle, inscribed in parabola. Tangents in points $D,E,F$ forms triangle $ABC$. Prove that $S_{DEF}=2S_{ABC}$. ($S_T$ is area of triangle $T$).
From F.S.Macaulay's book «Geometrical Conics», suggested by M. Panov
1 reply
NO_SQUARES
Yesterday at 9:08 AM
Luis González
4 hours ago
J
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FE solution too simple?
Yiyj1   9
N 4 hours ago by jasperE3
Source: 101 Algebra Problems from the AMSP
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the equality $$f(f(x)+y) = f(x^2-y)+4f(x)y$$holds for all pairs of real numbers $(x,y)$.

My solution

I feel like my solution is too simple. Is there something I did wrong or something I missed?
9 replies
Yiyj1
Apr 9, 2025
jasperE3
4 hours ago
J
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interesting function equation (fe) in IR
skellyrah   2
N 4 hours ago by jasperE3
Source: mine
find all function F: IR->IR such that $$ xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy) $$
2 replies
skellyrah
Yesterday at 9:51 AM
jasperE3
4 hours ago
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J
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One on octagon
Rushil   4
N Sep 24, 2019 by Mrmaths123
Source: Indian RMO 1992 Problem 9
The cyclic octagon $ABCDEFGH$ has sides $a,a,a,a,b,b,b,b$ respectively. Find the radius of the circle that circumscribes $ABCDEFGH.$
4 replies
Rushil
Oct 15, 2005
Mrmaths123
Sep 24, 2019
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One on octagon
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Reveal topic tags
geometry
circumcircle
symmetry
trigonometry
trig identities
Law of Cosines
L
G H BBookmark kLocked kLocked NReply
Source: Indian RMO 1992 Problem 9
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Rushil
1592 posts
Rushil
#1 h  Oct 15, 2005, 5:31 AM • 2 Y
Y by Adventure10, Mango247
The cyclic octagon $ABCDEFGH$ has sides $a,a,a,a,b,b,b,b$ respectively. Find the radius of the circle that circumscribes $ABCDEFGH.$
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frt
1294 posts
frt
#2 h  Oct 15, 2005, 3:36 PM • 4 Y
Y by Sayan, tantheta67, Adventure10, Mango247
Click to reveal hidden text
This post has been edited 1 time. Last edited by frt, Nov 19, 2005, 6:31 PM
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10000th User
3049 posts
10000th User
#3 h  Nov 19, 2005, 2:55 PM • 2 Y
Y by Adventure10, Mango247
frt: There is a mistake in your solution. Check cosine law again :ninja:
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frt
1294 posts
frt
#4 h  Nov 19, 2005, 6:33 PM • 2 Y
Y by Adventure10, Mango247
Thanks for pointing it out.

I edited my solution, so I hope the answer is correct now...
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Mrmaths123
1 post
Mrmaths123
#7 h  Sep 24, 2019, 4:38 AM • 3 Y
Y by Adventure10, Mango247, Paper123___--
We can reshape this octagon into another octagon of consecutive sides of a and b and inscribe it inside a square to find the radius
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