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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
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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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Inequality with condition a+b+c = ab+bc+ca (and special equality case)
DoThinh2001   69
N 35 minutes ago by mihaig
Source: BMO 2019, problem 2
Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$
Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases.

(Edit: Proposed by sir Leonard Giugiuc, Romania)
69 replies
DoThinh2001
May 2, 2019
mihaig
35 minutes ago
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1 line solution to Inequality
ItzsleepyXD   1
N 37 minutes ago by mihaig
Source: Own , Mock Thailand Mathematic Olympiad P8
Let $x_1,x_2,\dots,x_n$ be positive real integer such that $x_1^2+x_2^2+\cdots+x_n^2=2$ Prove that
$$\sum_{i=1}^{n}\frac{1}{x_i^3(x_{i-1}+x_{i+1})}\geqslant \left(\sum_{i=1}^{n}\frac{x_i}{x_{i-1}+x_{i+1}}\right)^3$$such that $x_{n+1}=x_1$ and $x_0=x_n$
1 reply
ItzsleepyXD
5 hours ago
mihaig
37 minutes ago
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a nice problem of nt from PUMaC
Namisgood   1
N 38 minutes ago by Goutamioqmtopper
Source: PUMaC
Problem is attached
1 reply
Namisgood
Yesterday at 9:47 AM
Goutamioqmtopper
38 minutes ago
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Diophantine equation !
ComplexPhi   11
N 40 minutes ago by Goutamioqmtopper
Determine all triples $(m , n , p)$ satisfying :
\[n^{2p}=m^2+n^2+p+1\]
where $m$ and $n$ are integers and $p$ is a prime number.
11 replies
ComplexPhi
Feb 4, 2015
Goutamioqmtopper
40 minutes ago
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Inequlities
sqing   33
N 42 minutes ago by sqing
Let $ a,b,c\geq 0 $ and $ a^2+ab+bc+ca=3 .$ Prove that$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2} \geq \frac{3}{2}$$$$\frac{1}{1+a^2}+ \frac{1}{1+b^2}+ \frac{1}{1+c^2}-bc \geq -\frac{3}{2}$$
33 replies
sqing
Jul 19, 2024
sqing
42 minutes ago
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Solve $\sin(17x)+\sin(13x)=\sin(7x)$
Speed2001   1
N an hour ago by rchokler
How to solve the equation:
$$ \sin(17x)+\sin(13x)=\sin(7x),\;0<x<24^{\circ} $$Approach: I'm trying to factor $\sin(18x)$ to get $x=10^{\circ}$.

Any hint would be appreciated.
1 reply
Speed2001
Yesterday at 1:06 AM
rchokler
an hour ago
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7^a - 3^b divides a^4 + b^2 (from IMO Shortlist 2007)
Dida Drogbier   39
N an hour ago by ND_
Source: ISL 2007, N1, VAIMO 2008, P4
Find all pairs of natural numbers $ (a, b)$ such that $ 7^a - 3^b$ divides $ a^4 + b^2$.

Author: Stephan Wagner, Austria
39 replies
Dida Drogbier
Apr 21, 2008
ND_
an hour ago
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Inspired by tom-nowy
sqing   0
an hour ago
Source: Own
Let $a,b,c \in [-\frac{1}{2}, \frac{1}{2}]$. Prove that
$$ \frac{1}{16}\geq (ab+bc+ca)^2+ a^2b^2c^2-(a+b+c)^2\geq -\frac{107}{64} $$$$ \frac{1}{16}\geq (ab+bc+ca)^2+3a^2b^2c^2-(a+b+c)^2\geq -\frac{105}{64} $$Let $a,b,c \in [-2,2]$. Prove that
$$172\geq (ab+bc+ca)^2+ a^2b^2c^2-(a+b+c)^2\geq -\frac{64}{11} $$$$300\geq (ab+bc+ca)^2+3a^2b^2c^2-(a+b+c)^2\geq -\frac{192}{35} $$
0 replies
sqing
an hour ago
0 replies
J
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An amazing inequality
Kei0923   6
N an hour ago by Kei0923
Source: Own (2021 Mock Japan MO Finals 5)
Let $n$ be a positive integer and $a_{1},a_{2},\dots, a_{n+1}$ be positive real numbers such that $$\displaystyle {\sum_{i=1}^{n+1}}\frac{1}{a_{i}+n^2}=\frac{1}{n}.$$Prove that $$\sum_{i=1}^{n+1} \frac{1}{a_{i}^{2021}+n^{2020}} \geq \frac{1}{n^{2020}}.$$
6 replies
Kei0923
Mar 14, 2021
Kei0923
an hour ago
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Prime Numbers
TRcrescent27   7
N an hour ago by Goutamioqmtopper
Source: 2015 Turkey JBMO TST
Let $p,q$ be prime numbers such that their sum isn't divisible by $3$. Find the all $(p,q,r,n)$ positive integer quadruples satisfy:
$$p+q=r(p-q)^n$$
Proposed by Şahin Emrah
7 replies
TRcrescent27
Jun 22, 2016
Goutamioqmtopper
an hour ago
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Very tasteful inequality
tom-nowy   1
N an hour ago by sqing
Let $a,b,c \in (-1,1)$. Prove that $$(a+b+c)^2+3>(ab+bc+ca)^2+3(abc)^2.$$
1 reply
tom-nowy
4 hours ago
sqing
an hour ago
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Inequalities
sqing   8
N an hour ago by sqing
Let $x\in(-1,1). $ Prove that
$$ \dfrac{1}{\sqrt{1-x^2}} + \dfrac{1}{2+ x^2} \geq \dfrac{3}{2}$$$$ \dfrac{2}{\sqrt{1-x^2}} + \dfrac{1}{1+x^2} \geq 3$$
8 replies
sqing
Apr 26, 2025
sqing
an hour ago
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Algebraic Manipulation
Darealzolt   0
an hour ago
Find the number of pairs of real numbers $a, b, c$ that satisfy the equation $a^4 + b^4 + c^4 + 1 = 4abc$.
0 replies
Darealzolt
an hour ago
0 replies
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Ugly Inequality
EthanWYX2009   1
N an hour ago by NTstrucker
Show that
\[\sum_{k=1}^n\frac{\phi(k)}k\ln\frac{2^k}{2^k-1}\ge 1-\frac 1{2^n}.\]
1 reply
1 viewing
EthanWYX2009
2 hours ago
NTstrucker
an hour ago
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ratio of any 2 sides >2/5 if 2h_b=h_a+h_c (2013 Kukin MO by OmSU 11.2)
parmenides51   0
Apr 12, 2021
One of the altitudes of the triangle is equal to the arithmetic mean of the other heights. Prove that the ratio of any two sides of this triangle is greater than $2/5$.

0 replies
parmenides51
Apr 12, 2021
0 replies
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ratio of any 2 sides >2/5 if 2h_b=h_a+h_c (2013 Kukin MO by OmSU 11.2)
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parmenides51
30650 posts
parmenides51
#1 h  Apr 12, 2021, 5:02 PM
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One of the altitudes of the triangle is equal to the arithmetic mean of the other heights. Prove that the ratio of any two sides of this triangle is greater than $2/5$.
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