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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
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0 replies
jlacosta
Mar 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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a,b,c,d shift inequality
verigo   6
N 3 minutes ago by SomeonecoolLovesMaths
Source: sweden
Let $a,b,c,d>0$ prove that
\[ \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{d^2} + \frac{d^2}{a^2}\geq\frac{a}{d} + \frac{b}{a} + \frac{c}{b} + \frac{d}{c}\]If you know same tasks, link them please
6 replies
verigo
Feb 14, 2018
SomeonecoolLovesMaths
3 minutes ago
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postaffteff
JetFire008   4
N 21 minutes ago by JetFire008
Source: Internet
Let $P$ be the Fermat point of a $\triangle ABC$. Prove that the Euler line of the triangles $PAB$, $PBC$, $PCA$ are concurrent and the point of concurrence is $G$, the centroid of $\triangle ABC$.
4 replies
JetFire008
38 minutes ago
JetFire008
21 minutes ago
J
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i need help
MR.1   3
N 22 minutes ago by MR.1
Source: help
can you guys tell me problems about fe in $R+$(i know $R$ well). i want to study so if you guys have some easy or normal problems please send me
3 replies
MR.1
3 hours ago
MR.1
22 minutes ago
J
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D1010 : How it is possible ?
Dattier   6
N 26 minutes ago by whwlqkd
Source: les dattes à Dattier
Is it true that$$\forall n \in \mathbb N^*, (24^n \times B \mod A) \mod 2 = 0 $$?

A=172840090421781518678763921675392141786000436658021921275090402437796947824966464426797102
59525308036470431210259590181720483369539690621515342820528633073982816814653666658107757108
67856720572225880311472925624694183944650261079955759251769111321319421445397848518597584590
900951222557860592579005088853698315463815905425095325508106272375728975

B=227564340154808184720778276049144229526648735475052708528935496537676518846805227119017278
70644188547893224843051453107076145465733981826429238937805270372241433808862604677609912285
67577953725945090125797351518670892779468968705801340068681556238850340398780828104506916965
606659768601942798676554332768254089685307970609932846902
6 replies
Dattier
Mar 10, 2025
whwlqkd
26 minutes ago
J
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Checkers on the board
Didier2   0
30 minutes ago
Source: Khamovniki 2023 - 2024 (group 10 - 1)
On all black cells of $8 \times 8$ chess board, except for the $4 \times 4$ central square, there is a checker. In one move, a checker can jump over adjacent checker (diagonally), and this adjacent checker (that was jumped over) is removed. Is it possible to come up with a sequence of moves, so that only one checker is left on the board?
0 replies
Didier2
30 minutes ago
0 replies
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Sharygin 2025 CR P18
Gengar_in_Galar   4
N 33 minutes ago by Atomix43
Source: Sharygin 2025
Let $ABCD$ be a quadrilateral such that the excircles $\omega_{1}$ and $\omega_{2}$ of triangles $ABC$ and $BCD$ touching their sides $AB$ and $BD$ respectively touch the extension of $BC$ at the same point $P$. The segment $AD$ meets $\omega_{2}$ at point $Q$, and the line $AD$ meets $\omega_{1}$ at $R$ and $S$. Prove that one of angles $RPQ$ and $SPQ$ is right
Proposed by: I.Kukharchuk
4 replies
Gengar_in_Galar
Mar 10, 2025
Atomix43
33 minutes ago
J
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D1014 : Intersection of set
Dattier   0
38 minutes ago
Source: les dattes à Dattier
Let $A=\{1,...,n\}$ with $\forall i\in A,B_i \subset A$ and $\forall i \in A, \text{card}(\bigcap \limits_{k=1,k\neq i}^n B_k)\geq 2$.

Is it true that $\bigcap \limits_{k=1}^n B_k \neq \emptyset$?
0 replies
Dattier
38 minutes ago
0 replies
J
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If a^2024, b^2024, c^2024--- triangle, then (a/2024), b, c also (or similar)
NO_SQUARES   1
N an hour ago by NO_SQUARES
Source: Kvant 2025 no. 1 M2827 and The XIX Southern Mathematical Tournament
It is known about positive numbers $a, b, c$ that it is possible to form a triangle from segments of length $a^{2024}, b^{2024}, c^{2024}$. Prove that it is possible to reduce one of the numbers $a, b, c$ by $2024$ times and obtain the numbers $a', b', c'$ so that segments with lengths $a', b', c'$ can also be formed into a triangle.
L. Shatunov
1 reply
NO_SQUARES
Yesterday at 3:18 PM
NO_SQUARES
an hour ago
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Inspired by Kazakhstan 2017
sqing   2
N an hour ago by Ash_the_Bash07
Source: Own
Let $a,b,c\ge \frac{1}{2}$ and $a^2+b^2+c^2=2. $ Prove that
$$\left(\frac{2}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{2}{a}-\frac{1}{b}+\frac{1}{c}\right)\ge \frac{8}{3}$$
2 replies
1 viewing
sqing
2 hours ago
Ash_the_Bash07
an hour ago
J
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About old Inequality
perfect_square   1
N an hour ago by sqing
Source: Arqady
This is: $a,b,c>0$ which satisfy $abc=1$
Prove that: $ \frac{a+b+c}{3} \ge \sqrt[10]{\frac{a^3+b^3+c^3}{3}}$
By $ uvw $ method, I can assum $b=c=x,a=\frac{1}{x^2}$
But I can't prove:
$ \frac{2x+\frac{1}{x^2}}{3} \ge \sqrt[10]{ \frac{2x^3+ \frac{1}{x^6}}{3}} $
Is there an another way?
1 reply
perfect_square
5 hours ago
sqing
an hour ago
J
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"Expanding" permutations
Ciobi_   5
N 2 hours ago by MathLover_ZJ
Source: Izho Day 2 Problem 6
$\indent$ For a positive integer $n$, let $S_n$ be the set of bijective functions from $\{1,2,\dots ,n\}$ to itself. For a pair of positive integers $(a,b)$ such that $1 \leq a <b \leq n$, and for a permutation $\sigma \in S_n$, we say the pair $(a,b)$ is expanding for $\sigma$ if $|\sigma (a)- \sigma(b)| \geq |a-b|$
$\indent$ (a) Is it true that for all integers $n > 1$, there exists $\sigma \in S_n$ so that the number of pairs $(a,b)$ that are expanding for permutation $\sigma$ is less than $1000n\sqrt n$ ?
$\indent$ (b) Does there exist a positive integer $n>1$ and a permutation $\sigma \in S_n$ so that the number of pairs $(a,b)$ that are expanding for the permutation $\sigma$ is less than $\frac{n\sqrt n}{1000}$?
5 replies
Ciobi_
Jan 15, 2025
MathLover_ZJ
2 hours ago
J
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Number theory
Ecrin_eren   0
2 hours ago
Show that there are no prime numbers satisfying the equation

(p + r)^q + (q + r)^p = (p + q)^r.

0 replies
Ecrin_eren
2 hours ago
0 replies
J
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Wait wasn&#039;t it the reciprocal in the paper?
Supercali   7
N 2 hours ago by kes0716
Source: India TST 2023 Day 2 P1
Let $\mathbb{Z}_{\ge 0}$ be the set of non-negative integers and $\mathbb{R}^+$ be the set of positive real numbers. Let $f: \mathbb{Z}_{\ge 0}^2 \rightarrow \mathbb{R}^+$ be a function such that $f(0, k) = 2^k$ and $f(k, 0) = 1$ for all integers $k \ge 0$, and $$f(m, n) = \frac{2f(m-1, n) \cdot f(m, n-1)}{f(m-1, n)+f(m, n-1)}$$for all integers $m, n \ge 1$. Prove that $f(99, 99)<1.99$.

Proposed by Navilarekallu Tejaswi
7 replies
Supercali
Jul 9, 2023
kes0716
2 hours ago
J
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Functional equation
Ecrin_eren   0
2 hours ago
Find all functions f:R R satisfying the equation

f(f(x)y) + f(x+ f(y)) = x f(y) + f(x+y)

for all x,y real numbers

0 replies
Ecrin_eren
2 hours ago
0 replies
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J
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Sum and product equal to 8
MS_Kekas   2
N Nov 27, 2024 by Rohit-2006
Source: Kyiv City MO 2024 Round 1, Problem 8.1
Find the number of positive integers for which the product of digits and the sum of digits are the same and equal to $8$.
2 replies
MS_Kekas
Jan 28, 2024
Rohit-2006
Nov 27, 2024
J
Sum and product equal to 8
G H J
Reveal topic tags
number theory
Digits
L
G H BBookmark kLocked kLocked NReply
Source: Kyiv City MO 2024 Round 1, Problem 8.1
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MS_Kekas
275 posts
MS_Kekas
#1 h  Jan 28, 2024, 9:12 PM
Y by
Find the number of positive integers for which the product of digits and the sum of digits are the same and equal to $8$.
This post has been edited 1 time. Last edited by MS_Kekas, Jan 28, 2024, 9:58 PM
Z K Y
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jasperE3
11091 posts
jasperE3
#2 h  Jan 30, 2024, 12:39 AM
Y by
MS_Kekas wrote:
Find the number of positive integers for which the product of digits and the sum of digits are the same and equal to $8$.

$8$
$4+2+1+1$
$2+2+2+1+1$
Z K Y
The post below has been deleted. Click to close.
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Rohit-2006
166 posts
Rohit-2006
#3 h  Nov 27, 2024, 7:06 AM
Y by
jasperE3 wrote:
MS_Kekas wrote:
Find the number of positive integers for which the product of digits and the sum of digits are the same and equal to $8$.

$8$
$4+2+1+1$
$2+2+2+1+1$

Total $23$
Z K Y
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a