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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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lcm and gcd FE
jasperE3   9
N 14 minutes ago by AshAuktober
Source: IMOC 2018 N2
Find all functions $f:\mathbb N\to\mathbb N$ satisfying
$$\operatorname{lcm}(f(x),y)\gcd(f(x),f(y))=f(x)f(f(y))$$for all $x,y\in\mathbb N$.
9 replies
jasperE3
Aug 17, 2021
AshAuktober
14 minutes ago
J
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A three-variable functional inequality on non-negative reals
Tintarn   9
N 35 minutes ago by ErTeeEs06
Source: Dutch TST 2024, 1.2
Find all functions $f:\mathbb{R}_{\ge 0} \to \mathbb{R}$ with
\[2x^3zf(z)+yf(y) \ge 3yz^2f(x)\]for all $x,y,z \in \mathbb{R}_{\ge 0}$.
9 replies
Tintarn
Jun 28, 2024
ErTeeEs06
35 minutes ago
J
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*pop* Noice FE
EmilXM   3
N an hour ago by jasperE3
Source: Own
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$, such that,
$$f((x+y)f(xy)-f(x))= x^2y+xy^2-f(x)$$for all $x,y \in \mathbb{R}$.
3 replies
EmilXM
Jul 4, 2021
jasperE3
an hour ago
J
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Troll number theory
kokcio   0
an hour ago
Let $p,q,r>2$ are primes, which satisfy:
$q^4+(q+4)(r^2-q^2)=p$
$q|p+5$
$\sqrt{\frac{q^4-1}{r^2-1}}$ is an integer
Find all possible values of $p$.
0 replies
kokcio
an hour ago
0 replies
J
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Find the range of a symmetric function
DinDean   0
an hour ago
For $a,b,c\in[0,1]$, assume $a+b+c=1$, compute the range of
\[f(a,b,c)=(1-a^2)^2+(1-b^2)^2+(1-c^2)^2.\]
0 replies
+2 w
DinDean
an hour ago
0 replies
J
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Beautiful problem
luutrongphuc   13
N an hour ago by ItzsleepyXD
(Phan Quang Tri) Let triangle $ABC$ be circumscribed about circle $(I)$, and let $H$ be the orthocenter of $\triangle ABC$. The circle $(I)$ touches line $BC$ at $D$. The tangent to the circle $(BHC)$ at $H$ meets $BC$ at $S$. Let $J$ be the midpoint of $HI$, and let the line $DJ$ meet $(I)$ again at $X$. The tangent to $(I)$ parallel to $BC$ meets the line $AX$ at $T$. Prove that $ST$ is tangent to $(I)$.
13 replies
luutrongphuc
Apr 4, 2025
ItzsleepyXD
an hour ago
J
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Easy Number Theory
VicKmath7   2
N an hour ago by zaidova
Source: Greece Archimedes Junior 2016
If $n$ is positive integer and $p, q, r$ are primes solve the system: $pqr=n$ and $(p+1)(q+1)r=n+138$
2 replies
VicKmath7
Mar 16, 2020
zaidova
an hour ago
J
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Linear recurrence fits with factorial finitely often
Assassino9931   1
N 2 hours ago by internationalnick123456
Source: Bulgaria Balkan MO TST 2025
Let $k\geq 3$ be an integer. The sequence $(a_n)_{n\geq 1}$ is defined via $a_1 = 1$, $a_2 = k$ and
\[ a_{n+2} = ka_{n+1} + a_n \]for any positive integer $n$. Prove that there are finitely many pairs $(m, \ell)$ of positive integers such that $a_m = \ell!$.
1 reply
Assassino9931
Yesterday at 10:25 PM
internationalnick123456
2 hours ago
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Thanks u!
Ruji2018252   5
N 2 hours ago by lbh_qys
Let $a^2+b^2+c^2-2a-4b-4c=7(a,b,c\in\mathbb{R})$
Find minimum $T=2a+3b+6c$
5 replies
Ruji2018252
Yesterday at 5:52 PM
lbh_qys
2 hours ago
J
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Quadratic residues in a given interval
cyshine   21
N 2 hours ago by cursed_tangent1434
Source: Brazilian Math Olympiad, Problem 2
Find the number of integers $ c$ such that $ -2007 \leq c \leq 2007$ and there exists an integer $ x$ such that $ x^2 + c$ is a multiple of $ 2^{2007}$.
21 replies
cyshine
Nov 2, 2007
cursed_tangent1434
2 hours ago
J
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Inspired by Ruji2018252
sqing   1
N 2 hours ago by lbh_qys
Source: Own
Let $ a,b,c $ be reals such that $ a^2+b^2+c^2-2a-4b-4c=7. $ Prove that
$$ -4\leq 2a+b+2c\leq 20$$$$5-4\sqrt 3\leq a+b+c\leq 5+4\sqrt 3$$$$ 11-4\sqrt {14}\leq a+2b+3c\leq 11+4\sqrt {14}$$
1 reply
sqing
6 hours ago
lbh_qys
2 hours ago
J
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F=(F^3+F^3)/9-2F^3
Yiyj1   1
N 2 hours ago by sp0rtman00000
Source: 101 Algebra Problems for the AMSP
Define the Fibonacci sequence $F_n$ as $$F_1=F_2=1, F_{n+1}+F_n=F_{n-1}$$fir $n \in \mathbb{N}$. Prove that $$F_{2n}=\dfrac{F_{2n+2}^3+F_{2n-2}^3}{9}-2F_{2n}^3$$for all $n \ge 2$.
1 reply
Yiyj1
4 hours ago
sp0rtman00000
2 hours ago
J
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Funky and nice NT FE
yaybanana   0
2 hours ago
Source: own
find all functions $f: \mathbb{Z}^* \rightarrow \mathbb{Z}^*$ ,s.t :

$f(n)+f(m) \mid n^2+nf(m)$

for all $m,n \in \mathbb{Z}^*$
0 replies
yaybanana
2 hours ago
0 replies
J
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4 lines concurrent
Zavyk09   3
N 2 hours ago by ItzsleepyXD
Source: Homework
Let $ABC$ be triangle with circumcenter $(O)$ and orthocenter $H$. $BH, CH$ intersect $(O)$ again at $K, L$ respectively. Lines through $H$ parallel to $AB, AC$ intersects $AC, AB$ at $E, F$ respectively. Point $D$ such that $HKDL$ is a parallelogram. Prove that lines $KE, LF$ and $AD$ are concurrent at a point on $OH$.
3 replies
Zavyk09
Yesterday at 11:51 AM
ItzsleepyXD
2 hours ago
J
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J
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A simple and nice inequality in a triangle.
Virgil Nicula   7
N Jun 26, 2016 by Virgil Nicula
Source: Own (?)
I don't meet this inequality until today ...

PP. Let $ABC$ be a triangle with the angled-bisectors $BE$ , $CF$ where $E\in AC$ , $F\in AB\ .$ Prove that $\boxed{\ EF^2\ \le\ BF\cdot CE\ }\ (*)\ .$
7 replies
Virgil Nicula
May 19, 2016
Virgil Nicula
Jun 26, 2016
J
A simple and nice inequality in a triangle.
G H J
Reveal topic tags
inequalities
geometry
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G H BBookmark kLocked kLocked NReply
Source: Own (?)
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Virgil Nicula
7054 posts
Virgil Nicula
#1 h  May 19, 2016, 4:52 PM • 2 Y
Y by Adventure10, Mango247
I don't meet this inequality until today ...

PP. Let $ABC$ be a triangle with the angled-bisectors $BE$ , $CF$ where $E\in AC$ , $F\in AB\ .$ Prove that $\boxed{\ EF^2\ \le\ BF\cdot CE\ }\ (*)\ .$
Z K Y
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arqady
30184 posts
arqady
#2 h  May 19, 2016, 7:33 PM • 2 Y
Y by Adventure10, Mango247
Indeed, it's $(b-c)^2a(a+b+c)\geq0$.
Z K Y
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aopser123
136 posts
aopser123
#3 h  May 19, 2016, 7:49 PM • 2 Y
Y by Adventure10, Mango247
arqady wrote:
Indeed, it's $(b-c)^2a(a+b+c)\geq0$.

How did you get this?
Z K Y
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Virgil Nicula
7054 posts
Virgil Nicula
#4 h  May 19, 2016, 8:38 PM • 2 Y
Y by Adventure10, Mango247
arqady wrote:
Indeed, it's $(b-c)^2a(a+b+c)\geq0$.

Yes, it is and my conclusion. See proposed problem P6 from here. Please, met you at least once this inequality ?! Thank you.
This post has been edited 2 times. Last edited by Virgil Nicula, May 20, 2016, 12:57 AM
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arqady
30184 posts
arqady
#5 h  May 19, 2016, 10:35 PM • 2 Y
Y by Adventure10, Mango247
No, I did not see it before.
Z K Y
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Virgil Nicula
7054 posts
Virgil Nicula
#6 h  May 19, 2016, 11:06 PM • 2 Y
Y by Adventure10, Mango247
Thank you. The your words say me that the previous inequality is own. Other interesting

inequality $:$ if $X$ , $Y$ are the projections of $E$ , $F$ on $BC\ ,$ then $XY\le \frac a2\cdot\left(\frac b{a+b}+\frac c{a+c}\right)\ .$
This post has been edited 2 times. Last edited by Virgil Nicula, Jun 26, 2016, 6:20 PM
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arqady
30184 posts
arqady
#9 h  Jun 26, 2016, 6:56 AM • 1 Y
Y by Adventure10
Are these projections to $BC$?
This post has been edited 1 time. Last edited by arqady, Jun 26, 2016, 6:56 AM
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Virgil Nicula
7054 posts
Virgil Nicula
#10 h  Jun 26, 2016, 6:24 PM • 1 Y
Y by Adventure10
Sorry! Yes, I edited now.
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