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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.
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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
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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
srnjbr   0
19 minutes ago
a^2+b^2+c^2+x^2+y^2=1. Find the maximum value of the expression (ax+by)^2+(bx+cy)^2
0 replies
1 viewing
srnjbr
19 minutes ago
0 replies
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divisibility
srnjbr   0
22 minutes ago
Find all natural numbers n such that there exists a natural number l such that for every m members of the natural numbers the number m+m^2+...m^l is divisible by n.
0 replies
srnjbr
22 minutes ago
0 replies
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Graph Theory
JetFire008   1
N 25 minutes ago by JetFire008
Prove that for any Hamiltonian cycle, if it contain edge $e$, then it must not contain edge $e'$.
1 reply
JetFire008
29 minutes ago
JetFire008
25 minutes ago
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Inequality and function
srnjbr   0
25 minutes ago
Find all f:R--R such that for all x,y, yf(x)+f(y)>=f(xy)
0 replies
srnjbr
25 minutes ago
0 replies
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Inspired by hunghd8
sqing   1
N 28 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ a+b+c\geq 2+abc . $ Prove that
$$a^2+b^2+c^2- abc\geq \frac{7}{4}$$$$a^2+b^2+c^2-2abc \geq 1$$$$a^2+b^2+c^2- \frac{1}{2}abc\geq \frac{31}{16}$$$$a^2+b^2+c^2- \frac{8}{5}abc\geq \frac{34}{25}$$
1 reply
sqing
42 minutes ago
sqing
28 minutes ago
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Assisted perpendicular chasing
sarjinius   2
N 29 minutes ago by chisa36
Source: Philippine Mathematical Olympiad 2025 P7
In acute triangle $ABC$ with circumcenter $O$ and orthocenter $H$, let $D$ be an arbitrary point on the circumcircle of triangle $ABC$ such that $D$ does not lie on line $OB$ and that line $OD$ is not parallel to line $BC$. Let $E$ be the point on the circumcircle of triangle $ABC$ such that $DE$ is perpendicular to $BC$, and let $F$ be the point on line $AC$ such that $FA = FE$. Let $P$ and $R$ be the points on the circumcircle of triangle $ABC$ such that $PE$ is a diameter, and $BH$ and $DR$ are parallel. Let $M$ be the midpoint of $DH$.
(a) Show that $AP$ and $BR$ are perpendicular.
(b) Show that $FM$ and $BM$ are perpendicular.
2 replies
sarjinius
Mar 9, 2025
chisa36
29 minutes ago
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Find min
hunghd8   4
N 32 minutes ago by imnotgoodatmathsorry
Let $a,b,c$ be nonnegative real numbers such that $ a+b+c\geq 2+abc $. Find min
$$P=a^2+b^2+c^2.$$
4 replies
1 viewing
hunghd8
5 hours ago
imnotgoodatmathsorry
32 minutes ago
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Prime for square numbers
giangtruong13   1
N an hour ago by shanelin-sigma
Source: City’s Specialized Math Examination
Given that $a,b$ are natural numbers satisfy that: $\frac{a^3}{a+b}$ and $\frac{b^3}{a+b}$ are prime numbers. Prove that $$a^2+3ab+3a+b+1$$is a perfect squared number
1 reply
giangtruong13
2 hours ago
shanelin-sigma
an hour ago
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Inspired by hunghd8
sqing   0
an hour ago
Source: Own
Let $ a,b,c\geq 0 $ and $ a+b+c\geq 2+abc . $ Prove that
$$a^2+b^2+c^2-\frac{1}{2}a^2b^2c^2\geq 2$$$$a^2+b^2+c^2-abc-\frac{1}{2}a^2b^2c^2\geq \frac{3}{2}$$$$a^2+b^2+c^2- \frac{19}{10}abc-\frac{1}{2}a^2b^2c^2\geq -\frac{12}{25}$$$$a^2+b^2+c^2- \frac{3}{2}abc-\frac{1}{2}a^2b^2c^2\geq \frac{17\sqrt{17}-71}{16}$$
0 replies
sqing
an hour ago
0 replies
J
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Interesting inequality
sqing   5
N 2 hours ago by sqing
Source: Own
Let $ a,b >0. $ Prove that
$$ \frac{1}{\frac{a}{a+b}+\frac{a}{2b}} +\frac{1}{\frac{b}{a+b}+\frac{1}{2}} +\frac{a}{2b} \geq \frac{5}{2} $$
5 replies
sqing
Feb 26, 2025
sqing
2 hours ago
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sum of divisors nt
Soupboy0   0
2 hours ago
Source: own
Let $\epsilon(n)$ denote the sum of the sum of the factors of all positive $\mathbb Z \le n$, for example, $\epsilon(5) $ is the sum of the factors of $5$ added to the sum of the factors of $4$ and so on until the sum of the factors of $1$, which would be $(1+5)+(1+2+4)+(1+3)+(1+2)+(1) = 21$. Let $M(n)$ denote $\sum_{i=1}^{n} n \pmod{i}$. Show that $\epsilon(n) + M(n) = n^2$ or find a counterexample
0 replies
Soupboy0
2 hours ago
0 replies
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euler-totient function
Laan   2
N 2 hours ago by Laan
Proof that there are infinitely many positive integers $n$ such that
$\varphi(n)<\varphi(n+1)<\varphi(n+2)$
2 replies
Laan
Today at 7:13 AM
Laan
2 hours ago
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2 var inquality
sqing   5
N 2 hours ago by sqing
Source: Own
Let $ a,b $ be nonnegative real numbers such that $ a^2+ab+b^2+a+b=1. $ Prove that
$$ (ab+1)(a+b)\leq \frac{ 20}{27} $$$$ (ab+1)(a+b-1)\leq - \frac{ 10}{27} $$Let $ a,b $ be nonnegative real numbers such that $ a^2+b^2+a+b=1. $ Prove that
$$ (ab+1)(a+b)\leq \frac{ 5\sqrt 3-7}{2} $$$$ (ab+1)(a+b-1)\leq 3\sqrt 3- \frac{ 11}{2} $$
5 replies
sqing
Yesterday at 3:00 PM
sqing
2 hours ago
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Is it fake? how can someone score 12 in AIME but can&#039;t qualify RMO
Bruce_wayne123   3
N 3 hours ago by Bruce_wayne123
Source: https://www.reddit.com/r/JEENEETards/comments/1jgduci/op_qualified_for_usamo/#lightbox
He also claims to have scored 93 percentile in JEEM maths another thing which makes it more doubtful and also he didn't got any letter from MAA
3 replies
Bruce_wayne123
3 hours ago
Bruce_wayne123
3 hours ago
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infinite set of points with integral distances from each other
parmenides51   0
Nov 9, 2022
Source: Mathcenter Contest / Oly - Thai Forum 2009 R1 p3 https://artofproblemsolving.com/community/c3196914_mathcenter_contest
Prove that for each $k$ points in the plane, no three collinear and having integral distances from each other. If we have an infinite set of points with integral distances from each other, then all points are collinear.

(Anonymous314)

PS. wording needs to be fixed , source
0 replies
parmenides51
Nov 9, 2022
0 replies
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infinite set of points with integral distances from each other
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Reveal topic tags
combinatorics
combinatorial geometry
points
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Source: Mathcenter Contest / Oly - Thai Forum 2009 R1 p3 https://artofproblemsolving.com/community/c3196914_mathcenter_contest
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parmenides51
30628 posts
parmenides51
#1 h  Nov 9, 2022, 4:35 AM
Y by
Prove that for each $k$ points in the plane, no three collinear and having integral distances from each other. If we have an infinite set of points with integral distances from each other, then all points are collinear.

(Anonymous314)

PS. wording needs to be fixed , source
This post has been edited 5 times. Last edited by parmenides51, Dec 5, 2022, 3:31 PM
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