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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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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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Integer
Fang-jh   14
N a minute ago by Speedysolver1
Source: Chinese TST
Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} + 3^{\phi(n)} + \cdots + n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} + \frac {1}{p_{2}} + \cdots + \frac {1}{p_{k}} + \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)
14 replies
Fang-jh
Apr 5, 2008
Speedysolver1
a minute ago
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Inequality with a,b,c,d
GeoMorocco   0
2 minutes ago
Source: Moroccan Training 2025
Let $ a,b,c,d$ positive real numbers such that $ a+b+c+d=3+\frac{1}{abcd}$ . Prove that :
$$ a^2+b^2+c^2+d^2+5abcd \geq 9 $$
0 replies
GeoMorocco
2 minutes ago
0 replies
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Interesting inequalities
sqing   1
N 13 minutes ago by sqing
Source: Own
Let $ a,b $ be real numbers . Prove that
$$-\frac{k+3}{8(k^2+3k+2)}\leq \frac{ab + a + b +k+2}{ (a^2+k)(b^2+k)} \leq \frac{1}{k+3} $$Where $ k\in N^+.$
1 reply
1 viewing
sqing
34 minutes ago
sqing
13 minutes ago
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Number Theory Chain!
JetFire008   23
N 19 minutes ago by Primeniyazidayi
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
23 replies
JetFire008
Apr 7, 2025
Primeniyazidayi
19 minutes ago
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Rational Number Theory
Olympiadium   11
N 22 minutes ago by whwlqkd
Source: KJMO 2022 P8
Find all pairs $(x, y)$ of rational numbers such that $$xy^2=x^2+2x-3$$
11 replies
Olympiadium
Oct 29, 2022
whwlqkd
22 minutes ago
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Functional Equation:
Omerking   1
N 35 minutes ago by Nuran2010
Find all functions $f:\mathbb {R}\rightarrow \mathbb {R}$ such that:
$$f(x^2f(x)+f(y))=(f(x))^3+y$$
1 reply
Omerking
41 minutes ago
Nuran2010
35 minutes ago
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Poly with sequence give infinitely many prime divisors
Assassino9931   3
N 42 minutes ago by DinDean
Source: Bulgaria National Olympiad 2025, Day 1, Problem 3
Let $P(x)$ be a non-constant monic polynomial with integer coefficients and let $a_1, a_2, \ldots$ be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence $b_n = P(n)^{a_n} + 1$.
3 replies
Assassino9931
Yesterday at 1:51 PM
DinDean
42 minutes ago
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Geometric progression
hzbrl   3
N an hour ago by pooh123
find all quadruples $(a, b, c, d)$ where they are integers and $a+b+c+d=65$, where $a, b, c, d$ are in a gp.
3 replies
hzbrl
Yesterday at 11:46 AM
pooh123
an hour ago
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Iterated FE on positive integers
MarkBcc168   62
N an hour ago by Jupiterballs
Source: ELMO 2020 P1
Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that $$f^{f^{f(x)}(y)}(z)=x+y+z+1$$for all $x,y,z \in \mathbb{N}$.

Proposed by William Wang.
62 replies
MarkBcc168
Jul 28, 2020
Jupiterballs
an hour ago
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Interesting inequalities
sqing   9
N an hour ago by sqing
Source: Own
Let $ a,b $ be real numbers . Prove that
$$-\frac{1}{12}\leq \frac{ab+a+b+1}{(a^2+3)(b^2+3)}\leq \frac{1}{4} $$$$-\frac{5}{96}\leq \frac{ab+a+b+2}{(a^2+4)(b^2+4)}\leq \frac{1}{5} $$$$-\frac{1}{16}\leq \frac{ab+a+b+1}{(a^2+4)(b^2+4)}\leq \frac{3+\sqrt 5}{32} $$$$-\frac{1}{18}\leq \frac{ab+a+b+3}{(a^2+3)(b^2+3)}\leq \frac{2+\sqrt[3] 2+\sqrt[3] {4}}{12} $$
9 replies
+1 w
sqing
Today at 4:41 AM
sqing
an hour ago
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Interesting inequalities
sqing   0
an hour ago
Source: Own
Let $ a,b $ be real numbers . Prove that
$$-\frac{7}{240}\leq \frac{ab + a + b +4}{ (a^2+6)(b^2+6)} \leq \frac{1}{7} $$$$-\frac{1}{24}\leq \frac{ab + a + b +1}{ (a^2+6)(b^2+6)} \leq \frac{14+ \sqrt 7}{72} $$$$-\frac{7}{192}\leq \frac{ab + a + b +2}{ (a^2+6)(b^2+6)} \leq \frac{2+ \sqrt 2}{32} $$
0 replies
sqing
an hour ago
0 replies
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4 lines concurrent
Zavyk09   0
2 hours ago
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 $OH$.
0 replies
Zavyk09
2 hours ago
0 replies
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Christmas special mock geometry olympiad 2020
Nari_Tom   1
N 2 hours ago by Nari_Tom
Let $\triangle ABC$ be a triangle with incenter $I$, and circumcircle $\omega$. Let the circle with diameter $AI$ intersects $\omega$ at $S$ and $A$. Let $M$ be the midpoint arc $BAC$ and let $MI$ intersect $\omega$ again at a point $T$. Let the circumcircle of $\triangle MIS$ intersect the circumcircle of $\triangle BIC$ at a point $X$. Let the tangent at $S$ to $\omega$ intersect $BC$ at a point $V$. Prove that $MX$ and $VT$ intersect on $\omega$.
1 reply
+1 w
Nari_Tom
2 hours ago
Nari_Tom
2 hours ago
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Prove that x1=x2=....=x2025
Rohit-2006   3
N 2 hours ago by kamatadu
Source: A mock
The real numbers $x_1,x_2,\cdots,x_{2025}$ satisfy,
$$x_1+x_2=2\bar{x_1}, x_2+x_3=2\bar{x_2},\cdots, x_{2025}+x_1=2\bar{x_{2025}}$$Where {$\bar{x_1},\cdots,\bar{x_{2025}}$} is a permutation of $x_1,x_2,\cdots,x_{2025}$. Prove that $x_1=x_2=\cdots=x_{2025}$
3 replies
Rohit-2006
Today at 5:22 AM
kamatadu
2 hours ago
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sum of lengths of sides and diagonals >= 2(2+\sqrt2), when (ABCD)=1
parmenides51   0
Jul 31, 2018
Source: Spanish Mathematical Olympiad 1997 P5
Prove that in every convex quadrilateral of area $1$, the sum of the lengths of the sides and diagonals is not smaller than $2(2+\sqrt2)$.
0 replies
parmenides51
Jul 31, 2018
0 replies
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sum of lengths of sides and diagonals >= 2(2+\sqrt2), when (ABCD)=1
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Reveal topic tags
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Source: Spanish Mathematical Olympiad 1997 P5
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parmenides51
30630 posts
parmenides51
#1 h  Jul 31, 2018, 9:57 AM • 2 Y
Y by Adventure10, Mango247
Prove that in every convex quadrilateral of area $1$, the sum of the lengths of the sides and diagonals is not smaller than $2(2+\sqrt2)$.
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