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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Wednesday at 3:18 PM
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
Wednesday at 3:18 PM
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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Excalibur Identity
jjsunpu   9
N 4 minutes ago by fruitmonster97
proof is below
9 replies
jjsunpu
Yesterday at 3:27 PM
fruitmonster97
4 minutes ago
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Problem 1
blug   1
N 19 minutes ago by sqing
Source: Polish Math Olympiad 2025 Finals P1
Find all $(a, b, c, d)\in \mathbb{R}$ satisfying
\[\begin{aligned} \begin{cases} a+b+c+d=0,\\ a^2+b^2+c^2+d^2=12,\\ abcd=-3.\\ \end{cases} \end{aligned}\]
1 reply
blug
39 minutes ago
sqing
19 minutes ago
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New geometry problem
titaniumfalcon   2
N 19 minutes ago by fruitmonster97
Post any solutions you have, with explanation or proof if possible, good luck!
2 replies
titaniumfalcon
Yesterday at 10:40 PM
fruitmonster97
19 minutes ago
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.problem.
Cobedangiu   2
N 19 minutes ago by Lankou
Find the integer coefficients after expanding Newton's binomial:
$$(\frac{3}{2}-\frac{2}{3}x^2)^n (n \in Z)$$
2 replies
Cobedangiu
Today at 6:20 AM
Lankou
19 minutes ago
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Product of first m primes
joybangla   6
N 22 minutes ago by megarnie
Source: European Mathematical Cup 2013, Junior Division, P1
For $m\in \mathbb{N}$ define $m?$ be the product of first $m$ primes. Determine if there exists positive integers $m,n$ with the following property :
\[ m?=n(n+1)(n+2)(n+3) \]

Proposed by Matko Ljulj
6 replies
joybangla
Jul 3, 2014
megarnie
22 minutes ago
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Inspired by Polish 2025
sqing   0
25 minutes ago
Source: Own
Let $ a,b,c,d $ be reals such that $ a+b+c+d =0 $ and $ a^2+b^2+c^2+d^2=12.$ Prove that$$-3\leq abcd\leq 9$$Let $ a,b,c,d $ be reals such that $ a+b+c+d =0 $ and $ abcd=-3.$ Prove that$$a^2+b^2+c^2+d^2 \geq 12$$Let $ a,b,c,d $ be reals such that $ a+b+c+d =0 $ and $ abcd=9.$ Prove that$$a^2+b^2+c^2+d^2 \geq 12$$
0 replies
sqing
25 minutes ago
0 replies
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Problem 4
blug   0
26 minutes ago
Source: Polish Math Olympiad 2025 Finals P4
A positive integer $n\geq 2$ and a set $S$ consisting of $2n$ disting positive integers smaller than $n^2$ are given. Prove that there exists a positive integer $r\in \{1, 2, ..., n\}$ that can be written in the form $r=a-b$, for $a, b\in \mathbb{S}$ in at least $3$ different ways.
0 replies
blug
26 minutes ago
0 replies
J
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Problem 3
blug   0
30 minutes ago
Source: Polish Math Olympiad 2025 Finals P3
Positive integer $k$ and $k$ colors are given. We will say that a set of $2k$ points on a plane is $colorful$, if it contains exactly 2 points of each color and if lines connecting every two points of the same color are pairwise distinct. Find, in terms of $k$ the least integer $n\geq 2$ such that: in every set of $nk$ points of a plane, no three of which are collinear, consisting of $n$ points of every color there exists a $colorful$ subset.
0 replies
blug
30 minutes ago
0 replies
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D1010 : How it is possible ?
Dattier   15
N 36 minutes ago by maxal
Source: les dattes à Dattier
Is it true that$$\forall n \in \mathbb N^*, (24^n \times B \mod A) \mod 2 = 0 $$?

A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975

B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902
15 replies
Dattier
Mar 10, 2025
maxal
36 minutes ago
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Inequalities
sqing   23
N 42 minutes ago by sqing
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that
$$a^2+b^2+ ab +21abc\leq\frac{512}{441}$$Equality holds when $a=b=\frac{38}{21},c=\frac{5}{214}.$
$$a^2+b^2+ ab +19abc\leq\frac{10648}{9747}$$Equality holds when $a=b=\frac{22}{57},c=\frac{13}{57}.$
$$a^2+b^2+ ab +22abc\leq\frac{15625}{13068}$$Equality holds when $a=b=\frac{25}{66},c=\frac{8}{33}.$
23 replies
sqing
Mar 26, 2025
sqing
42 minutes ago
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One of a,b,c,d is not greater than -1
WakeUp   5
N an hour ago by Nari_Tom
Source: Baltic Way 2002
Let $a,b,c,d$ be real numbers such that
\[a+b+c+d=-2\]
\[ab+ac+ad+bc+bd+cd=0\]
Prove that at least one of the numbers $a,b,c,d$ is not greater than $-1$.
5 replies
WakeUp
Nov 13, 2010
Nari_Tom
an hour ago
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a really nice polynomial problem
Etemadi   8
N 2 hours ago by amirhsz
Source: Iranian TST 2018, third exam day 1, problem 3
$n>1$ and distinct positive integers $a_1,a_2,\ldots,a_{n+1}$ are  given. Does there exist a polynomial $p(x)\in\Bbb{Z}[x]$ of degree  $\le n$ that satisfies the following conditions?
a. $\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1 $
b. $\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1 $

Proposed by Mojtaba Zare
8 replies
Etemadi
Apr 18, 2018
amirhsz
2 hours ago
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Japanese NT
pomodor_ap   1
N 2 hours ago by Tkn
Source: Japan TST 2024 p6
Find all quadruples $(a, b, c, d)$ of positive integers such that
$$2^a3^b + 4^c5^d = 2^b3^a + 4^d5^c.$$
1 reply
pomodor_ap
Oct 5, 2024
Tkn
2 hours ago
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The locus of P with supplementary angles condition
WakeUp   3
N 2 hours ago by Nari_Tom
Source: Baltic Way 2001
Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.
3 replies
WakeUp
Nov 17, 2010
Nari_Tom
2 hours ago
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right triangle with sides 1, r, r^2 (Chile 2009 L1 p1)
parmenides51   2
N May 26, 2019 by Barley
Consider a triangle whose sides measure $1, r$, and $r^2$.
Determine all the values of $r$ in such a way that the triangle is right.
2 replies
parmenides51
May 26, 2019
Barley
May 26, 2019
J
right triangle with sides 1, r, r^2 (Chile 2009 L1 p1)
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Reveal topic tags
geometry
right triangle
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parmenides51
30629 posts
parmenides51
#1 h  May 26, 2019, 6:28 PM • 2 Y
Y by Adventure10, Mango247
Consider a triangle whose sides measure $1, r$, and $r^2$.
Determine all the values of $r$ in such a way that the triangle is right.
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tauros
1099 posts
tauros
#2 h  May 26, 2019, 6:58 PM • 2 Y
Y by Adventure10, Mango247
solution

@below: $r$ is a real number, not necessarily integer.
This post has been edited 1 time. Last edited by tauros, May 26, 2019, 7:32 PM
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Barley
22 posts
Barley
#3 h  May 26, 2019, 7:10 PM • 2 Y
Y by Adventure10, Mango247
Note that pythagorean triples are in the form $2mn, m^2-n^2,m^2+n^2$.
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