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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:
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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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Hard number theory problem
Omid Hatami   16
N 13 minutes ago by quantam13
Source: Iran 2002
$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?
16 replies
Omid Hatami
Apr 9, 2004
quantam13
13 minutes ago
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hard..........
Noname23   1
N 22 minutes ago by Noname23
problem
1 reply
Noname23
6 hours ago
Noname23
22 minutes ago
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Maximize non-intersecting/perpendicular diagonals!
cjquines0   36
N 36 minutes ago by endless_abyss
Source: 2016 IMO Shortlist C5
Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.
36 replies
cjquines0
Jul 19, 2017
endless_abyss
36 minutes ago
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Nice function question
srnjbr   2
N an hour ago by pco
Find all functions f:R+--R+ such that for all a,b>0, f(af(b)+a)(f(bf(a))+a)=1
2 replies
srnjbr
Today at 4:28 AM
pco
an hour ago
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Inequality with real numbers
JK1603JK   2
N an hour ago by SunnyEvan
Source: unknown
Let a,b,c are real numbers. Prove that (a^3+b^3+c^3+3abc)^4+(a+b+c)^3(a+b-c)^3(-a+b+c)^3(a-b+c)^3>=0
2 replies
JK1603JK
5 hours ago
SunnyEvan
an hour ago
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Mathhhhh
mathbetter   10
N an hour ago by togrulhamidli2011
Three turtles are crawling along a straight road heading in the same
direction. "Two other turtles are behind me," says the first turtle. "One turtle is
behind me and one other is ahead," says the second. "Two turtles are ahead of me
and one other is behind," says the third turtle. How can this be possible?
10 replies
mathbetter
Mar 20, 2025
togrulhamidli2011
an hour ago
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SONG circle?
YaoAOPS   1
N 2 hours ago by bin_sherlo
Source: own?
Let triangle $ABC$ have incenter $I$ and intouch triangle $DEF$. Let the circumcircle of $ABC$ intersect $(AEF)$ at $S$ and have center $O$. Let $N$ be the midpoint of arc $BAC$ on the circumcircle. Suppose quadrilateral $SONG$ is cyclic such that $X = SN \cap OG$ lies on $BC$. Show that $\angle XGD = 90^\circ$.
1 reply
YaoAOPS
4 hours ago
bin_sherlo
2 hours ago
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A touching question on perpendicular lines
Tintarn   1
N 2 hours ago by Mathzeus1024
Source: Bundeswettbewerb Mathematik 2025, Round 1 - Problem 3
Let $k$ be a semicircle with diameter $AB$ and midpoint $M$. Let $P$ be a point on $k$ different from $A$ and $B$.

The circle $k_A$ touches $k$ in a point $C$, the segment $MA$ in a point $D$, and additionally the segment $MP$. The circle $k_B$ touches $k$ in a point $E$ and additionally the segments $MB$ and $MP$.

Show that the lines $AE$ and $CD$ are perpendicular.
1 reply
Tintarn
Mar 17, 2025
Mathzeus1024
2 hours ago
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Inequality with ordering
JustPostChinaTST   7
N 2 hours ago by AshAuktober
Source: 2021 China TST, Test 1, Day 1 P1
Given positive integers $m$ and $n$. Let $a_{i,j} ( 1 \le i \le m, 1 \le j \le n)$ be non-negative real numbers, such that
$$ a_{i,1} \ge a_{i,2} \ge \cdots \ge a_{i,n} \text{ and } a_{1,j} \ge a_{2,j} \ge \cdots \ge a_{m,j} $$holds for all $1 \le i \le m$ and $1 \le j \le n$. Denote
$$ X_{i,j}=a_{1,j}+\cdots+a_{i-1,j}+a_{i,j}+a_{i,j-1}+\cdots+a_{i,1},$$$$ Y_{i,j}=a_{m,j}+\cdots+a_{i+1,j}+a_{i,j}+a_{i,j+1}+\cdots+a_{i,n}.$$Prove that
$$ \prod_{i=1}^{m} \prod_{j=1}^{n} X_{i,j} \ge \prod_{i=1}^{m} \prod_{j=1}^{n} Y_{i,j}.$$
7 replies
JustPostChinaTST
Mar 17, 2021
AshAuktober
2 hours ago
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D1010 : How it is possible ?
Dattier   13
N 2 hours ago by Dattier
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
13 replies
Dattier
Mar 10, 2025
Dattier
2 hours ago
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Find min
hunghd8   8
N 2 hours ago by imnotgoodatmathsorry
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$
8 replies
hunghd8
Yesterday at 12:10 PM
imnotgoodatmathsorry
2 hours ago
J
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Inequality and function
srnjbr   5
N 3 hours ago by pco
Find all f:R--R such that for all x,y, yf(x)+f(y)>=f(xy)
5 replies
srnjbr
Yesterday at 4:26 PM
pco
3 hours ago
J
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Difficult factorization
Dakernew192   1
N 3 hours ago by Thursday
x^5-2x+6
1 reply
Dakernew192
Jan 8, 2024
Thursday
3 hours ago
J
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every point is colored red or blue
Sayan   8
N 3 hours ago by Mathworld314
Source: ISI(BS) 2005 #9
Suppose that to every point of the plane a colour, either red or blue, is associated.

(a) Show that if there is no equilateral triangle with all vertices of the same colour then there must exist three points $A,B$ and $C$ of the same colour such that $B$ is the midpoint of $AC$.

(b) Show that there must be an equilateral triangle with all vertices of the same colour.
8 replies
Sayan
Jun 23, 2012
Mathworld314
3 hours ago
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J
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Surprisingly low answer to the question what is the maximum
mshtand1   2
N Thursday at 6:05 PM by sarjinius
Source: Ukrainian Mathematical Olympiad 2025. Day 2, Problem 8.6, 10.5
Given $2025$ positive integer numbers such that the least common multiple (LCM) of all these numbers is not a perfect square. Mykhailo consecutively hides one of these numbers and writes down the LCM of the remaining $2024$ numbers that are not hidden. What is the maximum number of the $2025$ written numbers that can be perfect squares?

Proposed by Oleksii Masalitin
2 replies
mshtand1
Mar 13, 2025
sarjinius
Thursday at 6:05 PM
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Surprisingly low answer to the question what is the maximum
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Reveal topic tags
number theory
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G H BBookmark kLocked kLocked NReply
Source: Ukrainian Mathematical Olympiad 2025. Day 2, Problem 8.6, 10.5
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mshtand1
77 posts
mshtand1
#1 h  Mar 13, 2025, 11:54 PM
Y by
Given $2025$ positive integer numbers such that the least common multiple (LCM) of all these numbers is not a perfect square. Mykhailo consecutively hides one of these numbers and writes down the LCM of the remaining $2024$ numbers that are not hidden. What is the maximum number of the $2025$ written numbers that can be perfect squares?

Proposed by Oleksii Masalitin
Z K Y
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BR1F1SZ
518 posts
BR1F1SZ
#2 h  Mar 13, 2025, 11:58 PM
Y by
Surprisingly, the answer is $\boxed{1}$.
Z K Y
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sarjinius
239 posts
sarjinius
#3 h  Thursday at 6:05 PM
Y by
Let $a_1, a_2, \dots, a_{2025}$ be the $2025$ numbers and $A = lcm(a_1, \dots, a_{2025})$. Since $A$ is not a perfect square, there must exist a prime $p$ such that $\nu_p(A)$ is odd, implying $max\{\nu_p(a_i)\}$ is odd. WLOG let $a_1$ have the highest degree of $p$ (which is odd), so $2024$ out of the $2025$ written numbers must contain $a_1$, implying each of their $\nu_p$ is odd and hence not perfect squares. So only at most $1$ can be a perfect square, which can be satisfied by $(2, 1, 1, \dots, 1)$.
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