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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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[*]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
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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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Complex numbers should be easy
RenheMiResembleRice   2
N 2 minutes ago by RandomMathGuy500
Source: Wenjing Kong
I cant do the last part. :(
2 replies
RenheMiResembleRice
Yesterday at 8:32 AM
RandomMathGuy500
2 minutes ago
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Mathhhhh
mathbetter   13
N 24 minutes ago by ayeshabatool
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?
13 replies
mathbetter
Mar 20, 2025
ayeshabatool
24 minutes ago
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Integer FE
GreekIdiot   0
an hour ago
Let $\mathbb{N}$ denote the set of positive integers
Find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all $a,b \in \mathbb{N}$ it holds that $f(ab+f(b-1))|bf(a+b)f(3b-2+a)$
0 replies
GreekIdiot
an hour ago
0 replies
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Find interger root
Zuyong   2
N 2 hours ago by WallyWalrus
Source: ?
Find $(k,m)\in \mathbb{Z}$ satisfying $$9 k^4 + 30 k^3 + 44 k^2 m + 105 k^2 + 20 k m - 120 k + 36 m^2 + 80 m - 240=0$$
2 replies
Zuyong
Oct 24, 2024
WallyWalrus
2 hours ago
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Math Problem
hashbrown2009   1
N 3 hours ago by aidan0626
Show that the inequality
$$\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|} \le \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}$$holds for all real numbers $x_1,x_2,\dots,x_n$.

Proposed by Calvin Deng.

Do not copy solutions.
1 reply
hashbrown2009
3 hours ago
aidan0626
3 hours ago
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Simple inequality
sqing   56
N 3 hours ago by Tamam
Source: Shortlist BMO 2018, A1
Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that:

$$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$
56 replies
sqing
May 3, 2019
Tamam
3 hours ago
J
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stuck on a system of recurrence sequence
Nonecludiangeofan   2
N 3 hours ago by Nonecludiangeofan
Please guys help me solve this nasty problem that i've been stuck for the past month:
Let \( (a_n) \) and \( (b_n) \) be two sequences defined by:
\[ a_{n+1} = \frac{1 + a_n + a_n b_n}{b_n} \quad \text{and} \quad b_{n+1} = \frac{1 + b_n + a_n b_n}{a_n} \]for all \( n \ge 0 \), with initial values \( a_0 = 1 \) and \( b_0 = 2 \).

Prove that:
\[ a_{2024} < 5. \]
(btw am still not comfortable with system of recurrence sequences)
2 replies
Nonecludiangeofan
Thursday at 10:32 PM
Nonecludiangeofan
3 hours ago
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Numbers on a Board
Olympiadium   14
N 3 hours ago by deduck
Source: RMM 2021/4
Consider an integer \(n \ge 2\) and write the numbers \(1, 2, \ldots, n\) down on a board. A move consists in erasing any two numbers \(a\) and \(b\), then writing down the numbers \(a+b\) and \(\vert a-b \vert\) on the board, and then removing repetitions (e.g., if the board contained the numbers \(2, 5, 7, 8\), then one could choose the numbers \(a = 5\) and \(b = 7\), obtaining the board with numbers \(2, 8, 12\)). For all integers \(n \ge 2\), determine whether it is possible to be left with exactly two numbers on the board after a finite number of moves.

Proposed by China
14 replies
Olympiadium
Oct 14, 2021
deduck
3 hours ago
J
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A nice problem
hanzo.ei   1
N 3 hours ago by alexheinis

Given a nonzero real number \(a\) and a polynomial \(P(x)\) with real coefficients of degree \(n\) (\(n > 1\)) such that \(P(x)\) has no real roots. Prove that the polynomial
\[ Q(x) \;=\; P(x) \;+\; a\,P'(x) \;+\; a^2\,P''(x) \;+\; \dots \;+\; a^n\,P^{(n)}(x) \]has no real roots.
1 reply
hanzo.ei
5 hours ago
alexheinis
3 hours ago
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Dear Sqing: So Many Inequalities...
hashtagmath   24
N 3 hours ago by GreekIdiot
I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics? Thank you :)
24 replies
hashtagmath
Oct 30, 2024
GreekIdiot
3 hours ago
J
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interesting set problem
Dr.Poe98   1
N 3 hours ago by americancheeseburger4281
Source: Brazil Cono Sur TST 2024 - T3/P3
For a pair of integers $a$ and $b$, with $0<a<b<1000$, a set $S\subset \begin{Bmatrix}1,2,3,...,2024\end{Bmatrix}$ $escapes$ the pair $(a,b)$ if for any elements $s_1,s_2\in S$ we have $\left|s_1-s_2\right| \notin \begin{Bmatrix}a,b\end{Bmatrix}$. Let $f(a,b)$ be the greatest possible number of elements of a set that escapes the pair $(a,b)$. Find the maximum and minimum values of $f$.
1 reply
Dr.Poe98
Oct 21, 2024
americancheeseburger4281
3 hours ago
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Reflection lies on incircle
MP8148   5
N 3 hours ago by deraxenrovalo
Source: GOWACA Mock Geoly P3
In triangle $ABC$ with incircle $\omega$, let $I$ be the incenter and $D$ be the point where $\omega$ touches $\overline{BC}$. Let $S$ be the point on $(ABC)$ with $\angle ASI = 90^\circ$ and $H$ be the orthocenter of $\triangle BIC$, so that $Q \ne S$ on $\overline{HS}$ also satisfies $\angle AQI = 90^\circ$. Prove that $X$, the reflection of $I$ over the midpoint of $\overline{DQ}$, lies on $\omega$.
5 replies
MP8148
Aug 6, 2021
deraxenrovalo
3 hours ago
J
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Symmetric inequality FTW
Kimchiks926   20
N 4 hours ago by Marcus_Zhang
Source: Latvian TST for Baltic Way 2020 P1
Prove that for positive reals $a,b,c$ satisfying $a+b+c=3$ the following inequality holds:
$$ \frac{a}{1+2b^3}+\frac{b}{1+2c^3}+\frac{c}{1+2a^3} \ge 1 $$
20 replies
Kimchiks926
Oct 17, 2020
Marcus_Zhang
4 hours ago
J
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Interesting problem
V-217   0
4 hours ago
On the side $(BC)$ of the triangle $ABC$ consider a mobile point $M$. Let $B'$ the orthogonal projection of $B$ on $AM$. If the mobile points $N\in (BB'$ and $P\in (AM$ are such that $ANPC$ is a paralellogram, find the locus of point $P$ when $M$ goes through $BC$.
0 replies
V-217
4 hours ago
0 replies
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J
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Inequality
JK1603JK   2
N Mar 20, 2025 by lbh_qys
Source: unknown
Let a,b,c>=0 and ab+bc+ca>0 then prove \sqrt{a+b}+\sqrt{c+b}+\sqrt{a+c}\ge 2\sqrt[4]{ab+bc+ca}+\sqrt{\frac{a(b-c)^2+b(c-a)^2+c(a-b)^2}{ab+bc+ca}}
2 replies
JK1603JK
Mar 20, 2025
lbh_qys
Mar 20, 2025
J
Inequality
G H J
Reveal topic tags
inequalities
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Source: unknown
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JK1603JK
7 posts
JK1603JK
#1 h  Mar 20, 2025, 2:58 AM
Y by
Let a,b,c>=0 and ab+bc+ca>0 then prove \sqrt{a+b}+\sqrt{c+b}+\sqrt{a+c}\ge 2\sqrt[4]{ab+bc+ca}+\sqrt{\frac{a(b-c)^2+b(c-a)^2+c(a-b)^2}{ab+bc+ca}}
Z K Y
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CHESSR1DER
34 posts
CHESSR1DER
#2 h  Mar 20, 2025, 9:00 AM
Y by
you should add dollar sign before and after LaTeX code.
Inequality is $\sqrt{a+b}+\sqrt{c+b}+\sqrt{a+c}\geq 2\sqrt[4]{ab+bc+ca}+\sqrt{\frac{a(b-c)^2+b(c-a)^2+c(a-b)^2}{ab+bc+ca}}$
This post has been edited 1 time. Last edited by CHESSR1DER, Mar 20, 2025, 9:01 AM
Z K Y
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lbh_qys
425 posts
lbh_qys
#3 h  Mar 20, 2025, 9:26 AM • 1 Y
Y by kiyoras_2001
WLOG \(a\geq b\geq c\). By the AM-GM inequality,
\[ \sqrt{b+c} + \sqrt{a+c} \ge 2 \sqrt[4]{(b+c)(a+c)} = 2 \sqrt[4]{ab+bc+ca+c^2} \ge 2 \sqrt[4]{ab+bc+ca}. \]Furthermore,
\[ \frac{a(b-c)^2 + b(c-a)^2 + c(a-b)^2}{ab+bc+ca} = a+b+c - \frac{9abc}{ab+bc+ca} \le a+b+c - \frac{9abc}{3ab} = a+b+c - 3c \le a+b. \]Hence, the original inequality holds.
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