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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 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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Inequality
MathsII-enjoy   5
N 9 minutes ago by MathsII-enjoy
A interesting problem generalized :-D
5 replies
MathsII-enjoy
Saturday at 1:59 PM
MathsII-enjoy
9 minutes ago
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Number Theory
fasttrust_12-mn   8
N 12 minutes ago by ErTeeEs06
Source: Pan African Mathematics Olympiad P1
Find all positive intgers $a,b$ and $c$ such that $\frac{a+b}{a+c}=\frac{b+c}{b+a}$ and $ab+bc+ca$ is a prime number
8 replies
1 viewing
fasttrust_12-mn
Aug 15, 2024
ErTeeEs06
12 minutes ago
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My Unsolved FE on R+
ZeltaQN2008   4
N 15 minutes ago by mashumaro
Source: IDK
Give $a>0$. Find all funcitions $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for all any $x,y\in (0,\infty):$
$$f(xf(y)+a)=yf(x+y+a)$$
4 replies
ZeltaQN2008
2 hours ago
mashumaro
15 minutes ago
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Cyclic Non-homogeneous in 3 variables
Math-wiz   49
N 28 minutes ago by justaguy_69
Source: RMO 2019 P3
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that
$$\frac{a}{a^2+b^3+c^3}+\frac{b}{b^2+a^3+c^3}+\frac{c}{c^2+a^3+b^3}\leq\frac{1}{5abc}$$
49 replies
Math-wiz
Oct 20, 2019
justaguy_69
28 minutes ago
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Sum of squares in 1865
Twoisaprime   2
N Today at 3:53 AM by EmptyMachine
Source: 2024 CWMO P1
For positive integer $n$, note $S_n=1^{2024}+2^{2024}+ \cdots +n^{2024}$.
Prove that there exists infinitely many positive integers $n$, such that $S_n$ isn’t divisible by $1865$ but $S_{n+1}$ is divisible by $1865$
2 replies
Twoisaprime
Aug 6, 2024
EmptyMachine
Today at 3:53 AM
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2015 solutions for quotient function!
raxu   49
N Today at 1:26 AM by blueprimes
Source: TSTST 2015 Problem 5
Let $\varphi(n)$ denote the number of positive integers less than $n$ that are relatively prime to $n$. Prove that there exists a positive integer $m$ for which the equation $\varphi(n)=m$ has at least $2015$ solutions in $n$.

Proposed by Iurie Boreico
49 replies
raxu
Jun 26, 2015
blueprimes
Today at 1:26 AM
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Permutation with no two prefix sums dividing each other
Assassino9931   2
N Yesterday at 11:55 PM by Assassino9931
Source: Bulgaria Team Contest, March 2025, oVlad
Does there exist an infinite sequence of positive integers $a_1, a_2 \ldots$, such that every positive integer appears exactly once as a member of the sequence and $a_1 + a_2 + \cdots + a_i$ divides $a_1 + a_2 + \cdots + a_j$ if and only if $i=j$?
2 replies
Assassino9931
Yesterday at 11:21 PM
Assassino9931
Yesterday at 11:55 PM
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Divisibility NT
reni_wee   0
Yesterday at 8:35 PM
Source: Japan 1996, ONTCP
Let $m,n$ be relatively prime positive integers. Calculate $gcd(5^m+7^m, 5^n+7^n).$
0 replies
reni_wee
Yesterday at 8:35 PM
0 replies
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Modular arithmetic at mod n
electrovector   3
N Yesterday at 8:05 PM by Primeniyazidayi
Source: 2021 Turkey JBMO TST P6
Integers $a_1, a_2, \dots a_n$ are different at $\text{mod n}$. If $a_1, a_2-a_1, a_3-a_2, \dots a_n-a_{n-1}$ are also different at $\text{mod n}$, we call the ordered $n$-tuple $(a_1, a_2, \dots a_n)$ lucky. For which positive integers $n$, one can find a lucky $n$-tuple?
3 replies
electrovector
May 24, 2021
Primeniyazidayi
Yesterday at 8:05 PM
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Finding Solutions
MathStudent2002   22
N Yesterday at 6:03 PM by ihategeo_1969
Source: Shortlist 2016, Number Theory 5
Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\]Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.
22 replies
MathStudent2002
Jul 19, 2017
ihategeo_1969
Yesterday at 6:03 PM
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Floor sequence
va2010   87
N Yesterday at 4:53 PM by Mathgloggers
Source: 2015 ISL N1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \]contains at least one integer term.
87 replies
va2010
Jul 7, 2016
Mathgloggers
Yesterday at 4:53 PM
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INMO 2019 P3
div5252   45
N Yesterday at 4:51 PM by anudeep
Let $m,n$ be distinct positive integers. Prove that
$$gcd(m,n) + gcd(m+1,n+1) + gcd(m+2,n+2) \le 2|m-n| + 1. $$Further, determine when equality holds.
45 replies
div5252
Jan 20, 2019
anudeep
Yesterday at 4:51 PM
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Hard diophant equation
MuradSafarli   6
N Yesterday at 2:53 PM by iniffur
Find all positive integers $x, y, z, t$ such that the equation

$$ 2017^x + 6^y + 2^z = 2025^t $$
is satisfied.
6 replies
MuradSafarli
May 2, 2025
iniffur
Yesterday at 2:53 PM
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Average of elements is a perfect power
Valentin Vornicu   4
N Yesterday at 2:04 PM by Assassino9931
Source: Balkan MO 2000, problem 4
Show that for any $n$ we can find a set $X$ of $n$ distinct integers greater than 1, such that the average of the elements of any subset of $X$ is a square, cube or higher power.
4 replies
Valentin Vornicu
Apr 24, 2006
Assassino9931
Yesterday at 2:04 PM
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We can draw a diagonal
Amir Hossein   4
N Apr 26, 2013 by Saint123
Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$
4 replies
Amir Hossein
Sep 30, 2010
Saint123
Apr 26, 2013
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We can draw a diagonal
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Amir Hossein
5452 posts
Amir Hossein
#1 h  Sep 30, 2010, 1:21 PM • 2 Y
Y by Adventure10, Mango247
Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$
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TheChainheartMachine
1439 posts
TheChainheartMachine
#2 h  Apr 25, 2013, 1:28 AM • 1 Y
Y by Adventure10
Edit: What was I thinking? Was I even already fully awake when I posted this? Kyaaaah! *facepalm*
This post has been edited 1 time. Last edited by TheChainheartMachine, Apr 25, 2013, 6:45 AM
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chaotic_iak
2932 posts
chaotic_iak
#3 h  Apr 25, 2013, 6:17 AM • 3 Y
Y by TheChainheartMachine, Adventure10, Mango247
^ What? Drawing all the diagonals from a vertex cuts the hexagon to four triangles only.

(FYI: TheChainheartMachine originally posted a solution along the lines of "diagonals from a vertex cut a hexagon to six triangles", that is now edited out.)
This post has been edited 1 time. Last edited by chaotic_iak, Apr 25, 2013, 11:42 AM
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Domi
58 posts
Domi
#4 h  Apr 25, 2013, 10:05 AM • 2 Y
Y by Adventure10, Mango247
First suppose that the main diagonals pass through one point and use pigeon hole .

Domi
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Saint123
183 posts
Saint123
#5 h  Apr 26, 2013, 6:50 AM • 2 Y
Y by Adventure10, Mango247
Amir Hossein wrote:
Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$
It's obvious if the primary diagonals pass through one point, say O. Let the hexagon be $ABCDEI$.
By primary diagonals, I mean diagonals that do not directly cut off a triangle, like AD,BE and CI.
Let $\triangle EOD$ be the triangle with the required property. Now one of the triangles $\triangle ECD$ or $\triangle EID$ has area less that $\triangle EOD$ and we are done.

Now, let the primary diagonals make a triangle FGH, with F&G on IC, H on BE & AD.
Look at the triangles - $\triangle ABF,BFC,GDC,AHI,EGD,EHI$
From the preceding principle there exists one triangle satisfying the given requirement.
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