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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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Leibnitz theorem?
soryn   3
N 20 minutes ago by soryn
If M îs a interior point of the triangle ABC, and Ga,GB,GC are the centoids of triangles MBC, MAC and MAB, respectively, G0 is the centroid of triangle GaGbGc, show that the line MG0 passes through a fixed point.
3 replies
soryn
2 hours ago
soryn
20 minutes ago
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A hard inequality
JK1603JK   1
N 24 minutes ago by arqady
Source: unknown
Let $a,b,c\ge 0: a^2+b^2+c^2\le 3$ then prove $$a^3+b^3+c^3\ge a^3b^3+b^3c^3+c^3a^3.$$
1 reply
JK1603JK
Yesterday at 12:54 PM
arqady
24 minutes ago
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Prove that |a|≥2ⁿ+1
Rohit-2006   0
31 minutes ago
$P\in\mathbb{Z}[x]$ has degree $n$ having $n$ real roots in $(0,1)$. Suppose $a$ is the leading coefficient of $P$. Show that, $$\mid a\mid\geq2^n+1$$
0 replies
Rohit-2006
31 minutes ago
0 replies
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Why is the old one deleted?
EeEeRUT   9
N 42 minutes ago by Jupiterballs
Source: EGMO 2025 P1
For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$for all $1 \leqslant i \leqslant m-1$

Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.

Proposed by Paulius Aleknavičius, Lithuania
9 replies
EeEeRUT
Apr 16, 2025
Jupiterballs
42 minutes ago
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JEE Related ig?
mikkymini2   14
N an hour ago by mikkymini2
Hey everyone,

Just wanted to see if there are any other JEE aspirants on this forum currently prepping for it[mention year if you can]

I am actually entering 10th this year and have decided to try for it...So this year is just going to go in me strengthening my math (IOQM level (heard its enough till Mains part, so will start from there) for the problem solving part, and learn some topics from 11th and 12th as well)

It would be great to connect with others who are going through the same thing - share study strategies, tips, resources, discuss, and maybe even form study groups(not sure how to tho :maybe: ) and motivate each other ig?. :D
So yea, cya later
14 replies
mikkymini2
Apr 10, 2025
mikkymini2
an hour ago
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divisibility+factorial+exponent
britishprobe17   0
an hour ago
Source: KTOM Maret 2025
Determine all pairs of natural numbers $(m,n)$ that satisfy $2^{n!}+1|2^{m!}+19$
0 replies
britishprobe17
an hour ago
0 replies
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inequalities
pennypc123456789   4
N an hour ago by KhuongTrang
If $a,b,c$ are positive real numbers, then
$$ \frac{a + b}{a + 7b + c} + \dfrac{b + c}{b + 7c + a}+\dfrac{c + a}{c + 7a + b} \geq \dfrac{2}{3}$$
we can generalize this problem
4 replies
pennypc123456789
Yesterday at 1:53 PM
KhuongTrang
an hour ago
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2 var inquality
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $ a+ b+ab+a^2+b^2=5. $ Prove that
$$ (9+\sqrt{21} ) (a+b-2)(5- 2ab) \ge 10(a-1)(b-1)(a-b)$$$$ (9+\sqrt{21} ) (a+b-2)(3- ab) \ge6(a-1)(b-1)(a-b)$$
1 reply
sqing
3 hours ago
sqing
an hour ago
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Perfect Square Function
Miku3D   16
N 2 hours ago by MathLuis
Source: 2021 APMO P5
Determine all Functions $f:\mathbb{Z} \to \mathbb{Z}$ such that $f(f(a)-b)+bf(2a)$ is a perfect square for all integers $a$ and $b$.
16 replies
Miku3D
Jun 9, 2021
MathLuis
2 hours ago
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2 var inquality
sqing   4
N 2 hours ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $ a+ b+2ab=4 . $ Prove that
$$ 3(a+b-2)(2 - ab) \ge (a-1)(b-1)(a-b)$$$$ 9 (a+b-2)(3 - 2ab) \ge 2\sqrt 5(a-1)(b-1)(a-b)$$$$9(a+b-2)(6 - 5ab) \ge2\sqrt {14} (a-1)(b-1)(a-b)$$
4 replies
sqing
3 hours ago
sqing
2 hours ago
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Inspired by JK1603JK
sqing   2
N 2 hours ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ ab+bc+ca=3. $ Prove that
$$ (a+b+c-3)(12-5abc)\ge 2(a-b)(b-c)(c-a)$$$$6(a+b+c-3)(5-2abc)\ge 5(a-b)(b-c)(c-a)$$$$2(a+b+c-3)(9-5abc)\ge 3(a-b)(b-c)(c-a)$$$$3(a+b+c-3)(14-5abc)\ge 7(a-b)(b-c)(c-a)$$
2 replies
sqing
Yesterday at 2:11 PM
sqing
2 hours ago
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Inequalities
sqing   6
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
$$ \frac{3}{ 2}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }+ab \geq \frac{4}{15 }$$$$ \frac{14}{ 15}\geq \frac{1}{ a^2+3 }+ \frac{1}{ b^2+3 }-ab \geq -\frac{1}{2 }$$$$ \frac{3}{ 2}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }+ab \geq \frac{13}{42 }$$$$ \frac{41}{ 42}\geq \frac{1}{ a^4+3 }+ \frac{1}{ b^4+3 }-ab \geq -\frac{1}{2 }$$
6 replies
sqing
Apr 16, 2025
sqing
3 hours ago
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Inequalities
hn111009   1
N 3 hours ago by hn111009
Source: Maybe anywhere?
Let $a,b,c>0;r,s\in\mathbb{R}$ satisfied $a+b+c=1.$ Find minimum and maximum of $$P=a^rb^s+b^rc^s+c^ra^s.$$
1 reply
hn111009
Apr 13, 2025
hn111009
3 hours ago
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simplfy this
Miranda2829   6
N 5 hours ago by Miranda2829
4b+13/ -4b+15 = 1/6

anyone can help on the steps to do this ? thank u
6 replies
Miranda2829
Apr 16, 2025
Miranda2829
5 hours ago
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J
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2 square carpets in a square hall (Oral Moscow 2017 VI p3)
parmenides51   3
N Aug 29, 2020 by natmath
Two square carpets were brought to a large square hall, the side of one carpet is double than the side of the other. When they were placed in opposite corners of the hall, they cover a common area of $4$ m$^2$ , and when they were placed in adjacent corners, they cover a common area of $14$ m$^2$. What is the size of the hall?
3 replies
parmenides51
Aug 29, 2020
natmath
Aug 29, 2020
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2 square carpets in a square hall (Oral Moscow 2017 VI p3)
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Reveal topic tags
geometry
square
areas
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parmenides51
30629 posts
parmenides51
#1 h  Aug 29, 2020, 6:07 PM • 2 Y
Y by Mango247, Mango247
Two square carpets were brought to a large square hall, the side of one carpet is double than the side of the other. When they were placed in opposite corners of the hall, they cover a common area of $4$ m$^2$ , and when they were placed in adjacent corners, they cover a common area of $14$ m$^2$. What is the size of the hall?
This post has been edited 1 time. Last edited by parmenides51, Aug 29, 2020, 7:07 PM
Reason: added the word ''common area'', see #3
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natmath
8219 posts
natmath
#2 h  Aug 29, 2020, 6:55 PM
Y by
Shouldn't they cover less area when placed in adjacent corners?
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parmenides51
30629 posts
parmenides51
#3 h  Aug 29, 2020, 7:06 PM
Y by
first incorrect wording with original wording in Russian

those areas are the common areas covered by both carpets, I shall correct it
This post has been edited 1 time. Last edited by parmenides51, Aug 29, 2020, 7:08 PM
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natmath
8219 posts
natmath
#4 h  Aug 29, 2020, 7:12 PM
Y by
The smallest square has length $a$ and the biggest has length $b$
So the common area in the first case is a square of length 2, so $3a-2=b$. The common area in the first case is a 2x7 rectangle, so $a=7$. So $b=19$.

The hall is a 19x19 square with area 361
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