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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
J
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Interesting inequality
sqing   1
N 23 minutes ago by sqing
Source: Own
Let $ a,b,c\geq \frac{1}{3}$ and $ a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+8 . $ Prove that
$$ ab+bc +ca\leq 17+2\sqrt{73}$$Let $ a,b,c\geq \frac{1}{2}$ and $ a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+8 . $ Prove that
$$ ab+bc +ca\leq \frac{469+115\sqrt{17}}{32}$$Let $ a,b,c\geq \frac{1}{5}$ and $ a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+8 . $ Prove that
$$ ab+bc +ca\leq \frac{569+34\sqrt{281}}{25}$$
1 reply
sqing
43 minutes ago
sqing
23 minutes ago
J
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True Generalization of 2023 CGMO T7
EthanWYX2009   0
41 minutes ago
Source: aops.com/community/c6h3132846p28384612
Given positive integer $n.$ Let $x_1,\ldots ,x_n\ge 0$ and $x_1x_2\cdots x_n\le 1.$ Show that
\[\sum_{k=1}^n\frac{1}{1+\sum_{j\neq k}x_j}\le\frac n{1+(n-1)\sqrt[n]{x_1x_2\cdots x_n}}.\]
0 replies
EthanWYX2009
41 minutes ago
0 replies
J
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Not homogenous, messy inequality
Kimchiks926   10
N an hour ago by Marcus_Zhang
Source: Latvian TST for Baltic Way 2019 Problem 1
Prove that for all positive real numbers $a, b, c$ with $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} =1$ the following inequality holds:
$$3(ab+bc+ca)+\frac{9}{a+b+c} \le \frac{9abc}{a+b+c} + 2(a^2+b^2+c^2)+1$$
10 replies
Kimchiks926
May 29, 2020
Marcus_Zhang
an hour ago
J
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Interesting inequality
sqing   0
an hour ago
Source: Own
Let $ a,b,c\geq 2.$ Prove that
$$ (a+1)(b+1)(c +1)-\frac{9}{4}abc\leq 9$$$$ (a+1)(b+1)(c +1)-\frac{23}{10}abc\leq\frac{43}{5}$$
0 replies
sqing
an hour ago
0 replies
J
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USA 97 [1/(b^3+c^3+abc) + ... >= 1/(abc)]
Maverick   45
N 2 hours ago by Marcus_Zhang
Source: USAMO 1997/5; also: ineq E2.37 in Book: Inegalitati; Authors:L.Panaitopol,V. Bandila,M.Lascu
Prove that, for all positive real numbers $ a$, $ b$, $ c$, the inequality
\[ \frac {1}{a^3 + b^3 + abc} + \frac {1}{b^3 + c^3 + abc} + \frac {1}{c^3 + a^3 + abc} \leq \frac {1}{abc} \]
holds.
45 replies
Maverick
Sep 12, 2003
Marcus_Zhang
2 hours ago
J
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The prime inequality learning problem
orl   137
N 3 hours ago by Marcus_Zhang
Source: IMO 1995, Problem 2, Day 1, IMO Shortlist 1995, A1
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc = 1$. Prove that
\[ \frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}. \]
137 replies
orl
Nov 9, 2005
Marcus_Zhang
3 hours ago
J
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hard ............ (2)
Noname23   2
N 3 hours ago by mathprodigy2011
problem
2 replies
Noname23
Yesterday at 5:10 PM
mathprodigy2011
3 hours ago
J
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Abelkonkurransen 2025 3a
Lil_flip38   5
N 3 hours ago by ariopro1387
Source: abelkonkurransen
Let \(ABC\) be a triangle. Let \(E,F\) be the feet of the altitudes from \(B,C\) respectively. Let \(P,Q\) be the projections of \(B,C\) onto line \(EF\). Show that \(PE=QF\).
5 replies
Lil_flip38
Yesterday at 11:14 AM
ariopro1387
3 hours ago
J
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Inequality by Po-Ru Loh
v_Enhance   54
N 4 hours ago by Marcus_Zhang
Source: ELMO 2003 Problem 4
Let $x,y,z \ge 1$ be real numbers such that \[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \] Prove that \[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]
54 replies
v_Enhance
Dec 29, 2012
Marcus_Zhang
4 hours ago
J
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Problem 5
Functional_equation   14
N 4 hours ago by ali123456
Source: Azerbaijan third round 2020(JBMO Shortlist 2019 N6)
$a,b,c$ are non-negative integers.
Solve: $a!+5^b=7^c$

Proposed by Serbia
14 replies
Functional_equation
Jun 6, 2020
ali123456
4 hours ago
J
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a^12+3^b=1788^c
falantrng   6
N 4 hours ago by ali123456
Source: Azerbaijan NMO 2024. Junior P3
Find all the natural numbers $a, b, c$ satisfying the following equation:
$$a^{12} + 3^b = 1788^c$$.
6 replies
+1 w
falantrng
Jul 8, 2024
ali123456
4 hours ago
J
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stuck on a system of recurrence sequence
Nonecludiangeofan   0
5 hours ago
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)
0 replies
Nonecludiangeofan
5 hours ago
0 replies
J
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A huge group of children compare their heights
Tintarn   5
N 5 hours ago by InCtrl
Source: All-Russian MO 2024 9.8
$1000$ children, no two of the same height, lined up. Let us call a pair of different children $(a,b)$ good if between them there is no child whose height is greater than the height of one of $a$ and $b$, but less than the height of the other. What is the greatest number of good pairs that could be formed? (Here, $(a,b)$ and $(b,a)$ are considered the same pair.)
Proposed by I. Bogdanov
5 replies
Tintarn
Apr 22, 2024
InCtrl
5 hours ago
J
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Iran Inequality
mathmatecS   15
N 5 hours ago by Marcus_Zhang
Source: Iran 1998
When $x(\ge1),$ $y(\ge1),$ $z(\ge1)$ satisfy $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2,$ prove in equality.
$$\sqrt{x+y+z}\ge\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}$$
15 replies
mathmatecS
Jun 11, 2015
Marcus_Zhang
5 hours ago
J
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J
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Problem 4
den_thewhitelion   3
N Wednesday at 2:26 PM by DensSv
Source: Second Romanian JBMO TST 2016
We have a 4x4 board.All 1x1 squares are white.A move is changing colours of all squares of a 1x3 rectangle from black to white and from white to black.It is possible to make all the 1x1 squares black after several moves?
3 replies
den_thewhitelion
Jun 15, 2016
DensSv
Wednesday at 2:26 PM
J
Problem 4
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Reveal topic tags
combinatorics
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Source: Second Romanian JBMO TST 2016
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den_thewhitelion
262 posts
den_thewhitelion
#1 h  Jun 15, 2016, 6:50 AM • 2 Y
Y by Adventure10, Mango247
We have a 4x4 board.All 1x1 squares are white.A move is changing colours of all squares of a 1x3 rectangle from black to white and from white to black.It is possible to make all the 1x1 squares black after several moves?
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MilosMilicev
241 posts
MilosMilicev
#2 h  Oct 15, 2016, 11:28 AM • 2 Y
Y by Adventure10, Mango247
Look at the cells a1, b2, c3, d4, a4, d1. At the begining we have that they are white and in one move we change the colour of exactly one of those cells.It means that the number of moves is even. On the other hand, if we look at the cells b1, c2, a3, d3, b4, we obtain that number of moves is odd. Contradiction.
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MathDelicacy12
33 posts
MathDelicacy12
#3 h  Aug 23, 2019, 3:35 PM • 2 Y
Y by Adventure10, Mango247
den_thewhitelion wrote:
We have a 4x4 board.All 1x1 squares are white.A move is changing colours of all squares of a 1x3 rectangle from black to white and from white to black.It is possible to make all the 1x1 squares black after several moves?

Colour the board as shown below. Note that in each move an equal number of red, blue and green squares are changed. But there aren’t an equal number of red and green squares. So, it isn’t possible to make all the squares white. $\blacksquare$
Attachments:
This post has been edited 1 time. Last edited by MathDelicacy12, Aug 23, 2019, 3:35 PM
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DensSv
57 posts
DensSv
#4 h  Wednesday at 2:26 PM
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