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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.

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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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d(2025^{a_i}-1) divides a_{n+1}
navi_09220114   2
N 23 minutes ago by mickeymouse7133
Source: TASIMO 2025 Day 2 Problem 5
Let $a_n$ be a strictly increasing sequence of positive integers such that for all positive integers $n\ge 1$
\[d(2025^{a_n}-1)|a_{n+1}.\]Show that for any positive real number $c$ there is a positive integers $N_c$ such that $a_n>n^c$ for all $n\geq N_c$.

Note. Here $d(m)$ denotes the number of positive divisors of the positive integer $m$.
2 replies
navi_09220114
Monday at 11:51 AM
mickeymouse7133
23 minutes ago
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Funky function
TheUltimate123   22
N 42 minutes ago by jasperE3
Source: CJMO 2022/5 (https://aops.com/community/c594864h2791269p24548889)
Find all functions \(f:\mathbb R\to\mathbb R\) such that for all real numbers \(x\) and \(y\), \[f(f(xy)+y)=(x+1)f(y).\]
Proposed by novus677
22 replies
TheUltimate123
Mar 20, 2022
jasperE3
42 minutes ago
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R+ FE f(f(xy)+y)=(x+1)f(y)
jasperE3   0
43 minutes ago
Source: p24734470
Find all functions $f:\mathbb R^+\to\mathbb R^+$ such that for all positive real numbers $x$ and $y$:
$$f(f(xy)+y)=(x+1)f(y).$$edit oops sorry I misinterpreted that the original poster had solved it, solvable tho
0 replies
jasperE3
43 minutes ago
0 replies
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Inequality with x+y+z=1.
FrancoGiosefAG   1
N an hour ago by Blackbeam999
Let $x,y,z$ be positive real numbers such that $x+y+z=1$. Show that
\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]
1 reply
FrancoGiosefAG
4 hours ago
Blackbeam999
an hour ago
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Find all numbers
Rushil   10
N an hour ago by SomeonecoolLovesMaths
Source: Indian RMO 1994 Problem 3
Find all 6-digit numbers $a_1a_2a_3a_4a_5a_6$ formed by using the digits $1,2,3,4,5,6$ once each such that the number $a_1a_2a_2\ldots a_k$ is divisible by $k$ for $1 \leq k \leq 6$.
10 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
an hour ago
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Some number theory
EeEeRUT   3
N an hour ago by MathLuis
Source: Thailand MO 2025 P9
Let $p$ be an odd prime and $S = \{1,2,3,\dots, p\}$
Assume that $U: S \rightarrow S$ is a bijection and $B$ is an integer such that $$B\cdot U(U(a)) - a \: \text{ is a multiple of} \: p \: \text{for all} \: a \in S$$Show that $B^{\frac{p-1}{2}} -1$ is a multiple of $p$.
3 replies
EeEeRUT
May 14, 2025
MathLuis
an hour ago
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Gcd
Rushil   5
N 2 hours ago by SomeonecoolLovesMaths
Source: Indian RMO 1994 problem 5
Let $A$ be a set of $16$ positive integers with the property that the product of any two distinct members of $A$ will not exceed 1994. Show that there are numbers $a$ and $b$ in the set $A$ such that the gcd of $a$ and $b$ is greater than 1.
5 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
2 hours ago
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Solve the system
Rushil   20
N 2 hours ago by SomeonecoolLovesMaths
Source: 0
Solve the system of equations for real $x$ and $y$: \begin{eqnarray*} 5x \left( 1 + \frac{1}{x^2 + y^2}\right) &=& 12 \\ 5y \left( 1 - \frac{1}{x^2+y^2} \right) &=& 4 . \end{eqnarray*}
20 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
2 hours ago
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Angles made with the median
BBNoDollar   1
N 2 hours ago by Ianis
Determine the measures of the angles of triangle \(ABC\), knowing that the median \(BM\) makes an angle of \(30^\circ\) with side \(AB\) and an angle of \(15^\circ\) with side \(BC\).
1 reply
BBNoDollar
4 hours ago
Ianis
2 hours ago
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Find all rationals s.t..
Rushil   12
N 2 hours ago by SomeonecoolLovesMaths
Source: Indian RMO 1994 Problem 7
Find the number of rationals $\frac{m}{n}$ such that

(i) $0 < \frac{m}{n} < 1$;

(ii) $m$ and $n$ are relatively prime;

(iii) $mn = 25!$.
12 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
2 hours ago
J
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An inequality
Rushil   11
N 2 hours ago by SomeonecoolLovesMaths
Source: Indian RMO 1994 Problem 8
If $a,b,c$ are positive real numbers such that $a+b+c = 1$, prove that \[ (1+a)(1+b)(1+c) \geq 8 (1-a)(1-b)(1-c) . \]
11 replies
Rushil
Oct 25, 2005
SomeonecoolLovesMaths
2 hours ago
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Minimum moves to reach 25
lkason   0
3 hours ago
Source: Final of the XXI Polish Championship in Mathematical and Logical Games
Mateusz plays a game of erasing-writing on a large board. The board is initially empty.

In each move, he can either:
-- Write two numbers equal to $1$ on the board.
-- Erase two numbers equal to $n$ and write instead the numbers $n-1$ and $n+1$.

What is the minimal number of moves Mateusz needs to make for the number 25 to appear on the board?

Note: Numbers on the board retain their values; their digits cannot be combined or split.

Spoiler, answer:
Click to reveal hidden text
0 replies
lkason
3 hours ago
0 replies
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Reals to reals FE
a_507_bc   7
N 3 hours ago by jasperE3
Source: IMOC 2023 A2
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(f(x)+y)(x-f(y)) = f(x)^2-f(y^2).$$
7 replies
a_507_bc
Sep 9, 2023
jasperE3
3 hours ago
J
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Generic Real-valued FE
lucas3617   3
N 4 hours ago by jasperE3
$f: \mathbb{R} -> \mathbb{R}$, find all functions where $f(2x+f(2y-x))+f(-x)+f(y)=2f(x)+f(y-2x)+f(2y)$ for all $x$,$y \in \mathbb{R}$
3 replies
lucas3617
Apr 25, 2025
jasperE3
4 hours ago
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J
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a combinatorial geometry problem
xyz123456   5
N Apr 18, 2025 by Fishheadtailbody
$A_i\left(x_i{,}y_i\right){,}0\le x_i{,}y_i\le 1{,}1\le i\le 6.prove\ that:\exists \ 1\le i<j\le 6\ {,}\left|A_iA_j\right|\le \frac{\sqrt{13}}{6}$
5 replies
xyz123456
Mar 3, 2025
Fishheadtailbody
Apr 18, 2025
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a combinatorial geometry problem
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Reveal topic tags
combinatorics
combinatorial geometry
geometry
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xyz123456
48 posts
xyz123456
#1 h  Mar 3, 2025, 12:41 PM
Y by
$A_i\left(x_i{,}y_i\right){,}0\le x_i{,}y_i\le 1{,}1\le i\le 6.prove\ that:\exists \ 1\le i<j\le 6\ {,}\left|A_iA_j\right|\le \frac{\sqrt{13}}{6}$
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kiyoras_2001
678 posts
kiyoras_2001
#2 h  Mar 3, 2025, 3:07 PM
Y by
I rushed...
This post has been edited 3 times. Last edited by kiyoras_2001, Mar 3, 2025, 5:36 PM
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xyz123456
48 posts
xyz123456
#5 h  Mar 4, 2025, 12:16 PM
Y by
bump this
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Math-Problem-Solving
66 posts
Math-Problem-Solving
#6 h  Apr 1, 2025, 5:06 AM
Y by
xyz123456 wrote:
bump this

Do you have the solution? Then please post.
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xyz123456
48 posts
xyz123456
#7 h  Apr 18, 2025, 12:34 PM
Y by
Math-Problem-Solving wrote:
xyz123456 wrote:
bump this

Do you have the solution? Then please post.

I don't have the solution.
Z K Y
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Fishheadtailbody
8 posts
Fishheadtailbody
#8 h  Apr 18, 2025, 1:45 PM • 1 Y
Y by kiyoras_2001
I can sense that \frac{\sqrt{13}}{6} = \sqrt{(\frac{1}{2})^2+(\frac{1}{3})^2}.

For example, by setting the unit square into 2 by 3 (6 rectangles), if any of the two points are lying in the same smaller rectangle, we are done.
Sadly, pigeon principle does gives anything here so there should be a better way to count.
Click to reveal hidden text

Well an archive here said that n = 6 was proved by Graham, but it is not a simple one at least in 1991.
https://archive.org/details/unsolvedproblems0000crof/page/108/mode/2up
Click to reveal hidden text
This post has been edited 1 time. Last edited by Fishheadtailbody, Apr 18, 2025, 4:09 PM
Reason: Added the archive.
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