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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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Inequalities
sqing   5
N 2 hours ago by sqing
Let $ a,b,c $ be real numbers such that $ a^2+b^2+c^2=1. $ Prove that$$ |a-b|+|b-2c|+|c-3a|\leq 5$$$$|a-2b|+|b-3c|+|c-4a|\leq \sqrt{42}$$$$ |a-b|+|b-\frac{11}{10}c|+|c-a|\leq \frac{29}{10}$$
5 replies
sqing
Apr 22, 2025
sqing
2 hours ago
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Complex Numbers Question
franklin2013   3
N 2 hours ago by KSH31415
Hello everyone! This is one of my favorite complex numbers questions. Have fun!

$f(z)=z^{720}-z^{120}$. How many complex numbers $z$ are there such that $|z|=1$ and $f(z)$ is an integer.

Hint
3 replies
franklin2013
Apr 20, 2025
KSH31415
2 hours ago
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Inspired by Humberto_Filho
sqing   0
2 hours ago
Source: Own
Let $ a,b\geq 0 $ and $a + b \leq 2$. Prove that
$$\frac{a^2+1}{(( a+ b)^2+1)^2} \geq \frac{1}{25} $$$$\frac{(a^2+1)(b^2+1)}{((a+b)^2+1)^2} \geq \frac{4}{25} $$$$ \frac{a^2+1}{(( a+ 2b)^2+1)^2} \geq \frac{1}{289} $$$$ \frac{a^2+1}{((2a+ b)^2+1)^2} \geq \frac{5}{289} $$


0 replies
sqing
2 hours ago
0 replies
J
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Inequalities
Scientist10   2
N 2 hours ago by arqady
If $x, y, z \in \mathbb{R}$, then prove that the following inequality holds:
\[ \sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x \]
2 replies
Scientist10
Yesterday at 6:36 PM
arqady
2 hours ago
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$n$ with $2000$ divisors divides $2^n+1$ (IMO 2000)
Valentin Vornicu   65
N 2 hours ago by ray66
Source: IMO 2000, Problem 5, IMO Shortlist 2000, Problem N3
Does there exist a positive integer $ n$ such that $ n$ has exactly 2000 prime divisors and $ n$ divides $ 2^n + 1$?
65 replies
Valentin Vornicu
Oct 24, 2005
ray66
2 hours ago
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Find the smallest of sum of elements
hlminh   0
2 hours ago
Let $S=\{1,2,...,2014\}$ and $X=\{a_1,a_2,...,a_{30}\}$ is a subset of $S$ such that if $a,b\in X,a+b\leq 2014$ then $a+b\in X.$ Find the smallest of $\dfrac{a_1+a_2+\cdots+a_{30}}{30}.$
0 replies
hlminh
2 hours ago
0 replies
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Easy IMO 2023 NT
799786   133
N 2 hours ago by Maximilian113
Source: IMO 2023 P1
Determine all composite integers $n>1$ that satisfy the following property: if $d_1$, $d_2$, $\ldots$, $d_k$ are all the positive divisors of $n$ with $1 = d_1 < d_2 < \cdots < d_k = n$, then $d_i$ divides $d_{i+1} + d_{i+2}$ for every $1 \leq i \leq k - 2$.
133 replies
799786
Jul 8, 2023
Maximilian113
2 hours ago
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Complicated FE
XAN4   2
N 3 hours ago by cazanova19921
Source: own
Find all solutions for the functional equation $f(xyz)+\sum_{cyc}f(\frac{yz}x)=f(x)\cdot f(y)\cdot f(z)$, in which $f$: $\mathbb R^+\rightarrow\mathbb R^+$
Note: the solution is actually quite obvious - $f(x)=x^n+\frac1{x^n}$, but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.
2 replies
XAN4
Yesterday at 11:53 AM
cazanova19921
3 hours ago
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Inequalities
sqing   28
N 3 hours ago by sqing
Let $ a,b $ be reals such that $ a^2+ab+b^2 =3$ . Prove that
$$ \frac{4}{ 3}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{11}{4 }$$$$ \frac{13}{ 4}\geq \frac{1}{ a^2+5 }+ \frac{1}{ b^2+5 }+ab \geq -\frac{2}{3 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq -\frac{17}{6 }$$$$ \frac{19}{ 6}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2}$$Let $ a,b $ be reals such that $ a^2-ab+b^2 =1 $ . Prove that
$$ \frac{3}{ 2}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }+ab \geq \frac{4}{15 }$$$$ \frac{14}{ 15}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }-ab \geq -\frac{1}{2 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq \frac{13}{42 }$$$$ \frac{41}{ 42}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2 }$$
28 replies
sqing
Apr 16, 2025
sqing
3 hours ago
J
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Cute diophantine
TestX01   0
3 hours ago
Find all sequences of four consecutive integers such that twice their product is perfect square minus nine.
0 replies
TestX01
3 hours ago
0 replies
J
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\frac{1}{9}+\frac{1}{\sqrt{3}}\geq a^2+\sqrt{a+ b^2} \geq \frac{1}{4}
sqing   1
N 3 hours ago by sqing
Source: Own
Let $a,b\geq 0 $ and $3a+4b =1 .$ Prove that
$$\frac{2}{3}\geq a +\sqrt{a^2+ 4b^2}\geq \frac{6}{13}$$$$\frac{1}{9}+\frac{1}{\sqrt{3}}\geq a^2+\sqrt{a+ b^2} \geq \frac{1}{4}$$$$2\geq a+\sqrt{a^2+16b} \geq \frac{2}{3}\geq a+\sqrt{a^2+16b^3} \geq \frac{2(725-8\sqrt{259})}{729}$$
1 reply
sqing
Oct 3, 2023
sqing
3 hours ago
J
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Stronger inequality than an old result
KhuongTrang   22
N 4 hours ago by KhuongTrang
Source: own, inspired
Problem. Find the best constant $k$ satisfying $$(ab+bc+ca)\left[\frac{1}{(a+b)^{2}}+\frac{1}{(b+c)^{2}}+\frac{1}{(c+a)^{2}}\right]\ge \frac{9}{4}+k\cdot\frac{a(a-b)(a-c)+b(b-a)(b-c)+c(c-a)(c-b)}{(a+b+c)^{3}}$$holds for all $a,b,c\ge 0: ab+bc+ca>0.$
22 replies
KhuongTrang
Aug 1, 2024
KhuongTrang
4 hours ago
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Something nice
KhuongTrang   26
N 4 hours ago by KhuongTrang
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
26 replies
KhuongTrang
Nov 1, 2023
KhuongTrang
4 hours ago
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can anyone solve this
averageguy   8
N 5 hours ago by Bole
Hi guys,
For some reason I can't think of a simple way to solve this problem. Is there anyway you guys can think of without trig or if it does have trig something elegant. Answer is 106 btw.
8 replies
averageguy
Dec 26, 2024
Bole
5 hours ago
J
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J
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Cyclic Penta Geo
DarthKitKat   1
N Jan 1, 2021 by parmenides51
If a regular pentagon is inscribed in a circle (VWXYZ) and P is on arc VW, how would you use Ptolemy's to prove VX+VZ=VW+VZ+VY
1 reply
DarthKitKat
Jan 3, 2018
parmenides51
Jan 1, 2021
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Cyclic Penta Geo
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Reveal topic tags
pentagon
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DarthKitKat
6 posts
DarthKitKat
#1 h  Jan 3, 2018, 2:06 AM • 1 Y
Y by Adventure10
If a regular pentagon is inscribed in a circle (VWXYZ) and P is on arc VW, how would you use Ptolemy's to prove VX+VZ=VW+VZ+VY
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parmenides51
30630 posts
parmenides51
#2 h  Jan 1, 2021, 12:16 PM • 1 Y
Y by Mango247
solved here
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