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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
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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Inequalities
idomybest   3
N an hour ago by damyan
Source: The Interesting Around Technical Analysis Three Variable Inequalities
The problem is in the attachment below.
3 replies
idomybest
Oct 15, 2021
damyan
an hour ago
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Function on positive integers with two inputs
Assassino9931   2
N 2 hours ago by Assassino9931
Source: Bulgaria Winter Competition 2025 Problem 10.4
The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$. Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$. Determine all possible values of $f(2025, 2025)$.
2 replies
Assassino9931
Jan 27, 2025
Assassino9931
2 hours ago
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Normal but good inequality
giangtruong13   4
N 2 hours ago by IceyCold
Source: From a province
Let $a,b,c> 0$ satisfy that $a+b+c=3abc$. Prove that: $$\sum_{cyc} \frac{ab}{3c+ab+abc} \geq \frac{3}{5} $$
4 replies
giangtruong13
Mar 31, 2025
IceyCold
2 hours ago
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inequalities of elements in set
toanrathay   0
2 hours ago
Let \( m \) be a positive integer such that \( m \geq 4 \), and let the set
\[ A = \{a_1, a_2, a_3, \ldots, a_m\} \]consist of distinct positive integers not exceeding 2025. Suppose that for every \( a, b \in A \), with \( a \ne b \), if \( a + b \leq 2025 \), then \( a + b \in A \) as well. Prove that

\[ \frac{a_1 + a_2 + a_3 + \cdots + a_m}{m} \geq 1013. \]
0 replies
toanrathay
2 hours ago
0 replies
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Tiling rectangle with smaller rectangles.
MarkBcc168   61
N 2 hours ago by YaoAOPS
Source: IMO Shortlist 2017 C1
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.

Proposed by Jeck Lim, Singapore
61 replies
MarkBcc168
Jul 10, 2018
YaoAOPS
2 hours ago
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A magician has one hundred cards numbered 1 to 100
Valentin Vornicu   49
N 2 hours ago by YaoAOPS
Source: IMO 2000, Problem 4, IMO Shortlist 2000, C1
A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn.

How many ways are there to put the cards in the three boxes so that the trick works?
49 replies
Valentin Vornicu
Oct 24, 2005
YaoAOPS
2 hours ago
J
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Nice inequality
sqing   2
N 2 hours ago by Seungjun_Lee
Source: WYX
Let $a_1,a_2,\cdots,a_n (n\ge 2)$ be real numbers . Prove that : There exist positive integer $k\in \{1,2,\cdots,n\}$ such that $$\sum_{i=1}^{n}\{kx_i\}(1-\{kx_i\})<\frac{n-1}{6}.$$Where $\{x\}=x-\left \lfloor x \right \rfloor.$
2 replies
sqing
Apr 24, 2019
Seungjun_Lee
2 hours ago
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Concurrency
Dadgarnia   27
N 2 hours ago by zuat.e
Source: Iranian TST 2020, second exam day 2, problem 4
Let $ABC$ be an isosceles triangle ($AB=AC$) with incenter $I$. Circle $\omega$ passes through $C$ and $I$ and is tangent to $AI$. $\omega$ intersects $AC$ and circumcircle of $ABC$ at $Q$ and $D$, respectively. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $CQ$. Prove that $AD$, $MN$ and $BC$ are concurrent.

Proposed by Alireza Dadgarnia
27 replies
Dadgarnia
Mar 12, 2020
zuat.e
2 hours ago
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nice geo
Melid   1
N 2 hours ago by Melid
Source: 2025 Japan Junior MO preliminary P9
Let ABCD be a cyclic quadrilateral, which is AB=7 and BC=6. Let E be a point on segment CD so that BE=9. Line BE and AD intersect at F. Suppose that A, D, and F lie in order. If AF=11 and DF:DE=7:6, find the length of segment CD.
1 reply
Melid
2 hours ago
Melid
2 hours ago
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product of all integers of form i^3+1 is a perfect square
AlastorMoody   3
N 2 hours ago by Assassino9931
Source: Balkan MO ShortList 2009 N3
Determine all integers $1 \le m, 1 \le n \le 2009$, for which
\begin{align*} \prod_{i=1}^n \left( i^3 +1 \right) = m^2 \end{align*}
3 replies
AlastorMoody
Apr 6, 2020
Assassino9931
2 hours ago
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IMO ShortList 1998, combinatorics theory problem 1
orl   44
N 3 hours ago by YaoAOPS
Source: IMO ShortList 1998, combinatorics theory problem 1
A rectangular array of numbers is given. In each row and each column, the sum of all numbers is an integer. Prove that each nonintegral number $x$ in the array can be changed into either $\lceil x\rceil $ or $\lfloor x\rfloor $ so that the row-sums and column-sums remain unchanged. (Note that $\lceil x\rceil $ is the least integer greater than or equal to $x$, while $\lfloor x\rfloor $ is the greatest integer less than or equal to $x$.)
44 replies
orl
Oct 22, 2004
YaoAOPS
3 hours ago
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Complex Numbers Question
franklin2013   2
N 4 hours ago by Xx_BABAI_xX
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
2 replies
franklin2013
Apr 20, 2025
Xx_BABAI_xX
4 hours ago
J
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Inequalities
sqing   17
N 4 hours ago by sqing
Let $ a,b,c> 0 $ and $ ab+bc+ca\leq 3abc . $ Prove that
$$ a+ b^2+c\leq a^2+ b^3+c^2 $$$$ a+ b^{11}+c\leq a^2+ b^{12}+c^2 $$
17 replies
sqing
Yesterday at 1:54 PM
sqing
4 hours ago
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How many ways can we indistribute n different marbles into 6 identical boxes
Taiharward   9
N 4 hours ago by mathprodigy2011
How many ways can we distribute n indifferent marbles into 6 identical boxes and one jar?
9 replies
1 viewing
Taiharward
Today at 2:14 AM
mathprodigy2011
4 hours ago
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Number of solutions
Ecrin_eren   4
N Apr 18, 2025 by Ecrin_eren
The given equation is:

x³ + 4y³ + 2y = (2024 + 2y)(xy + 1)

The question asks for the number of integer solutions.

4 replies
Ecrin_eren
Apr 3, 2025
Ecrin_eren
Apr 18, 2025
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Number of solutions
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Reveal topic tags
number theory
integer polynomials
Simon favorite factoring trick
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Ecrin_eren
56 posts
Ecrin_eren
#1 h  Apr 3, 2025, 11:27 AM
Y by
The given equation is:

x³ + 4y³ + 2y = (2024 + 2y)(xy + 1)

The question asks for the number of integer solutions.
Z K Y
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jjsunpu
130 posts
jjsunpu
#2 h  Apr 3, 2025, 3:54 PM
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for x or y?
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Ecrin_eren
56 posts
Ecrin_eren
#3 h  Apr 3, 2025, 6:41 PM
Y by
Number of (x,y) pairs
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rchokler
2966 posts
rchokler
#5 h  Apr 3, 2025, 8:00 PM
Y by
$x^3+4y^3+2y=(2024+2y)(xy+1)\implies x^3-2xy^2+4y^3-2024xy-2024=0\implies$

Putting $z=\frac{x}{y}$ for $y\neq 0$ gives $z^3-2z+4-\frac{2024z}{y}-\frac{1}{y^3}=0$ Letting $|y|\to\infty$ results in $z^3-2z+4=0\implies(z+2)(z^2-2z+2)=0\implies z=-2$.

So we have an asymptote $x+2y=c$.

So plug in $x=c-2y$ to get $(c-2y)^3+4y^3+2y=(2024+2y)((c-2y)y + 1)\implies (10c+4048)y^2-(6c^2+2024c)y+c^3-2024=0$.

Therefore, $y=\frac{3c^2+1012c\pm \sqrt{(3c^2+1012c)^2-(10c+4048)(c^3-2024)}}{10c+4048}=\frac{3c^2+1012c\pm\sqrt{-c^4 + 2024 c^3 + 1024144 c^2 + 20240 c + 8193152}}{10c+4048}$.

The integer range is $-419\leq c\leq 2443$. The discriminant is never a square.

There are no integer solutions for $x^3+4y^3+2y=(2024+2y)(xy+1)$.
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Ecrin_eren
56 posts
Ecrin_eren
#6 h  Apr 18, 2025, 3:19 PM
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Any other solutions ?
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