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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 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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JEE Related ig?
mikkymini2   10
N an hour ago by Idiot_of_the64squares
Hey everyone,

Just wanted to see if there are any other JEE aspirants on this forum currently prepping for it[mention year if you can]

I am actually entering 10th this year and have decided to try for it...So this year is just going to go in me strengthening my math (IOQM level (heard its enough till Mains part, so will start from there) for the problem solving part, and learn some topics from 11th and 12th as well)

It would be great to connect with others who are going through the same thing - share study strategies, tips, resources, discuss, and maybe even form study groups(not sure how to tho :maybe: ) and motivate each other ig?. :D
So yea, cya later
10 replies
mikkymini2
Thursday at 2:54 PM
Idiot_of_the64squares
an hour ago
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Inspired by old results
sqing   2
N an hour ago by sqing
Source: Own
Let $ a,b\in [0,1] $ . Prove that
$$(a+b)(\frac{1}{a+1}+\frac{k}{b+1})\leq k+1 $$Where $ k\geq 0. $
$$(a+b-ab)(\frac{1}{a+1}+\frac{k}{b+1})\leq \frac{2k+1}{2} $$Where $ k\geq1. $
2 replies
sqing
2 hours ago
sqing
an hour ago
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prove that any quadrilateral satisfying this inequality is a trapezoid
mqoi_KOLA   0
an hour ago
Prove that any quadrilateral satisfying this inequality is a Trapezoid/trapzium $$ |r - p| < q + s < r + p $$where $p,r$ are lengths of parallel sides and $q,s$ are other two sides.
0 replies
mqoi_KOLA
an hour ago
0 replies
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Sequence with non-positive terms
socrates   8
N 2 hours ago by ray66
Source: Baltic Way 2014, Problem 2
Let $a_0, a_1, . . . , a_N$ be real numbers satisfying $a_0 = a_N = 0$ and \[a_{i+1} - 2a_i + a_{i-1} = a^2_i\] for $i = 1, 2, . . . , N - 1.$ Prove that $a_i\leq 0$ for $i = 1, 2, . . . , N- 1.$
8 replies
socrates
Nov 11, 2014
ray66
2 hours ago
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Inequalities
sqing   0
2 hours ago
Let $ a,b,c\in [0,1] $ . Prove that
$$(a+b+c)\left(\frac{1}{a^2+3}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{19}{4}$$$$(a+b+c)\left(\frac{1}{a^2+ 4}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{23}{5}$$$$(a+b+c)\left(\frac{1}{a^2+ \frac{5}{2}}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{34}{7}$$$$(a+b+c)\left(\frac{1}{a^2+ \frac{7}{2}}+\frac{2}{b^2+2}+\frac{2}{c^2+2}\right)\leq \frac{14}{3}$$
0 replies
sqing
2 hours ago
0 replies
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Radii Relaionship
steveshaff   0
2 hours ago
Two externally tangent circles with radii a and b are each internally tangent to a semicircle and its diameter. The two points of tangency on the semicircle and the two points of tangency on its diameter lie on a circle of radius r. Prove that r^2 = 3ab.
0 replies
steveshaff
2 hours ago
0 replies
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lcm(1,2,3,...,n)
lgx57   5
N 2 hours ago by Kempu33334
Let $M=\operatorname{lcm}(1,2,3,\cdots,n)$.Estimate the range of $M$.
5 replies
lgx57
Apr 9, 2025
Kempu33334
2 hours ago
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Letters in grid
buzzychaoz   6
N 2 hours ago by de-Kirschbaum
Source: CGMO 2016 Q6
Find the greatest positive integer $m$, such that one of the $4$ letters $C,G,M,O$ can be placed in each cell of a table with $m$ rows and $8$ columns, and has the following property: For any two distinct rows in the table, there exists at most one column, such that the entries of these two rows in such a column are the same letter.
6 replies
buzzychaoz
Aug 14, 2016
de-Kirschbaum
2 hours ago
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Equal pairs in continuous function
CeuAzul   16
N 2 hours ago by Ilikeminecraft
Let $f(x)$ be an continuous function defined in $\text{[0,2015]},f(0)=f(2015)$
Prove that there exists at least $2015$ pairs of $(x,y)$ such that $f(x)=f(y),x-y \in \mathbb{N^+}$
16 replies
1 viewing
CeuAzul
Aug 6, 2018
Ilikeminecraft
2 hours ago
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Inequality
math2000   7
N 2 hours ago by imnotgoodatmathsorry
Let $a,b,c>0$.Prove that $\dfrac{1}{(a+b)\sqrt{(a+2c)(b+2c)}}>\dfrac{3}{2(a+b+c)^2}$
7 replies
math2000
Jan 22, 2021
imnotgoodatmathsorry
2 hours ago
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Thanks u!
Ruji2018252   9
N 3 hours ago by sqing
Let $a^2+b^2+c^2-2a-4b-4c=7(a,b,c\in\mathbb{R})$
Find minimum $T=2a+3b+6c$
9 replies
Ruji2018252
Apr 9, 2025
sqing
3 hours ago
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R+ Functional Equation
Mathdreams   7
N 3 hours ago by jasperE3
Source: Nepal TST 2025, Problem 3
Find all functions $f : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that \[f(f(x)) + xf(xy) = x + f(y)\]for all positive real numbers $x$ and $y$.

(Andrew Brahms, USA)
7 replies
Mathdreams
Yesterday at 1:27 PM
jasperE3
3 hours ago
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Incenter and midpoint geom
sarjinius   88
N 3 hours ago by LHE96
Source: 2024 IMO Problem 4
Let $ABC$ be a triangle with $AB < AC < BC$. Let the incenter and incircle of triangle $ABC$ be $I$ and $\omega$, respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$. Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$. Let $AI$ intersect the circumcircle of triangle $ABC$ at $P \ne A$. Let $K$ and $L$ be the midpoints of $AC$ and $AB$, respectively.
Prove that $\angle KIL + \angle YPX = 180^{\circ}$.

Proposed by Dominik Burek, Poland
88 replies
sarjinius
Jul 17, 2024
LHE96
3 hours ago
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ineq.trig.
wer   19
N 3 hours ago by mpcnotnpc
If a, b, c are the sides of a triangle, show that: $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\frac{r}{R}\le2$
19 replies
wer
Jul 5, 2014
mpcnotnpc
3 hours ago
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An inequality
JK1603JK   3
N Mar 31, 2025 by lbh_qys
Let a,b,c\ge 0: ab+bc+ca>0 then prove \frac{5ab+c^2}{a+b}+\frac{5bc+a^2}{b+c}+\frac{5ca+b^2}{c+a}\ge 9\cdot\frac{ab+bc+ca}{a+b+c}.
3 replies
JK1603JK
Mar 29, 2025
lbh_qys
Mar 31, 2025
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An inequality
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Reveal topic tags
inequalities
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JK1603JK
46 posts
JK1603JK
#1 h  Mar 29, 2025, 1:05 PM
Y by
Let a,b,c\ge 0: ab+bc+ca>0 then prove \frac{5ab+c^2}{a+b}+\frac{5bc+a^2}{b+c}+\frac{5ca+b^2}{c+a}\ge 9\cdot\frac{ab+bc+ca}{a+b+c}.
Z K Y
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vsarg
251 posts
vsarg
#2 h  Mar 29, 2025, 6:25 PM
Y by
Nooo noo formating hurt my eyes Jek1603k
Let $a,b,c\ge 0: ab+bc+ca>0$ then prove $\frac{5ab+c^2}{a+b}+\frac{5bc+a^2}{b+c}+\frac{5ca+b^2}{c+a}\ge 9\cdot\frac{ab+bc+ca}{a+b+c}.$
this much better. And I pr0bably will not see this again so welcome. If you don't say thank you I will take back you are welcome. I don't want to be that guy who says welcome before other guy says thank you. In Kentucky we are humble.
Your welcome
Z K Y
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Demetri
1249 posts
Demetri
#3 h  Mar 29, 2025, 6:31 PM
Y by
vsarg wrote:
Nooo noo formating hurt my eyes Jek1603k
Let $a,b,c\ge 0: ab+bc+ca>0$ then prove $\frac{5ab+c^2}{a+b}+\frac{5bc+a^2}{b+c}+\frac{5ca+b^2}{c+a}\ge 9\cdot\frac{ab+bc+ca}{a+b+c}.$
this much better. And I pr0bably will not see this again so welcome. If you don't say thank you I will take back you are welcome. I don't want to be that guy who says welcome before other guy says thank you. In Kentucky we are humble.
Your welcome
They can't latex......
Z K Y
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lbh_qys
507 posts
lbh_qys
#4 h  Mar 31, 2025, 3:46 AM
Y by
Multiplying both sides of the inequality by \(a+b+c\), we have

\[ (a+b+c)\frac{a^2 + 5bc}{b+c} = \frac{a(a-b)(a-c)}{b+c} + \frac{4abc}{b+c} + 2a^2 + 5bc, \]
so it is equivalent to proving

\[ \sum \frac{a(a-b)(a-c)}{b+c} + 4abc \sum \frac{1}{b+c} + 2\sum a^2 + 5\sum bc \geq 9 \sum bc. \]
According to Schur's inequality,

\[ \sum \frac{a(a-b)(a-c)}{b+c} = \frac{1}{(a+b)(b+c)(c+a)} \sum a(a^2-b^2)(a^2-c^2) \geq 0, \]
so it suffices to prove

\[ 4abc \sum \frac{1}{b+c} \geq 4 \sum bc - 2 \sum a^2. \]
This clearly holds by applying the Cauchy–Schwarz inequality,

\[ \sum \frac{1}{b+c} \geq \frac{9}{\sum (b+c)} = \frac{9}{2(a+b+c)}, \]
and Schur's inequality

\[ \frac{9abc}{a+b+c} \geq 2\sum bc - \sum a^2. \]
Hence, the original inequality is established.
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