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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 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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Another triangle
Rushil   14
N a minute ago by SomeonecoolLovesMaths
Source: Indian RMO 1991 Problem 1
Let $P$ be an interior point of a triangle $ABC$ and $AP,BP,CP$ meet the sides $BC,CA,AB$ in $D,E,F$ respectively. Show that \[ \frac{AP}{PD} = \frac{AF}{FB} + \frac{AE}{EC}. \]
Remark
14 replies
1 viewing
Rushil
Oct 15, 2005
SomeonecoolLovesMaths
a minute ago
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13rd ibmo - rep. dominicana 1998/q2.
carlosbr   6
N 8 minutes ago by fearsum_fyz
Source: Spanish Communities
The circumference inscribed on the triangle $ABC$ is tangent to the sides $BC$, $CA$ and $AB$ on the points $D$, $E$ and $F$, respectively. $AD$ intersect the circumference on the point $Q$. Show that the line $EQ$ meet the segment $AF$ at its midpoint if and only if $AC=BC$.
6 replies
carlosbr
Apr 16, 2006
fearsum_fyz
8 minutes ago
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Inspired by old results
sqing   1
N an hour ago by sqing
Source: Own
Let $ a, b\geq 0, a+b=2. $ Prove that
$$\frac {24}{25} < \frac {1}{a^3 + 1 + ab}+\frac {1}{b^3 +1 + ab} +\frac {1}{a^3 + b^3 + ab} \leq \frac {89}{72}$$Let $ a, b\geq 0, a+b+ab=3. $ Prove that
$$\frac {4}{5} < \frac {1}{a^3 + 1 + ab}+\frac {1}{b^3 +1 + ab} +\frac {1}{a^3 + b^3 + ab} \leq \frac {811}{756}$$Let $ a, b\geq 0, a+b+ab=2. $ Prove that
$$\frac {23}{20} < \frac {1}{a^3 + 1 + ab}+\frac {1}{b^3 +1 + ab} +\frac {1}{a^3 + b^3 + ab} \leq \frac {404+291\sqrt{3}}{506}$$
1 reply
sqing
2 hours ago
sqing
an hour ago
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RMM 2013 Problem 6
dr_Civot   15
N an hour ago by N3bula
A token is placed at each vertex of a regular $2n$-gon. A move consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.
15 replies
dr_Civot
Mar 3, 2013
N3bula
an hour ago
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3-variable inequality with min(ab,bc,ca)>=1
mathwizard888   72
N 2 hours ago by math-olympiad-clown
Source: 2016 IMO Shortlist A1
Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$
Proposed by Tigran Margaryan, Armenia
72 replies
mathwizard888
Jul 19, 2017
math-olympiad-clown
2 hours ago
J
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Hard Inequality
JARP091   1
N 2 hours ago by lbh_qys
Source: Own?
Let \( a, b, c > 0 \) with \( abc = 1 \). Prove that
\[ \frac{a^5}{b^2 + 2c^3} + \frac{2b^5}{3c + a^6} + \frac{c^7}{a + b^4} \geq 2. \]
1 reply
JARP091
5 hours ago
lbh_qys
2 hours ago
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USAMO solutions
ABCD1728   1
N 2 hours ago by ohiorizzler1434
Do USAMO USATSTST and USATST have OFFICIAL solutions? Thanks!
1 reply
ABCD1728
3 hours ago
ohiorizzler1434
2 hours ago
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Distance between any two points is irrational
orl   21
N 3 hours ago by cursed_tangent1434
Source: IMO 1987, Day 2, Problem 5
Let $n\ge3$ be an integer. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
21 replies
orl
Nov 11, 2005
cursed_tangent1434
3 hours ago
J
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Symmetric inequality
mrrobotbcmc   5
N 3 hours ago by youochange
Let a,b,c,d belong to positive real numbers such that a+b+c+d=1. Prove that a^3/(b+c)+b^3/(c+d)+c^3/(d+a)+d^3/(a+b)>=1/8
5 replies
mrrobotbcmc
5 hours ago
youochange
3 hours ago
J
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Two perpendiculars
jayme   0
3 hours ago
Source: Own?
Dear Mathlinkers,

1. ABC a triangle
2. 0 the circumcircle
3. D the pole of BC wrt 0
4. B', C' the symmetrics of B, C wrt AC, AB
5. 1b, 1c the circumcircles of the triangles BB'D, CC'D
6. J the center of 1b
7. V the second point of intersection of DJ and 1c.

Prove : CV is perpendicular to BC.

Sincerely
Jean-Louis
0 replies
1 viewing
jayme
3 hours ago
0 replies
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Probably a good lemma
Zavyk09   4
N 4 hours ago by Zavyk09
Source: found when solving exercises
Let $ABC$ be a triangle with circumcircle $\omega$. Arbitrary points $E, F$ on $AC, AB$ respectively. Circumcircle $\Omega$ of triangle $AEF$ intersects $\omega$ at $P \ne A$. $BE$ intersects $CF$ at $I$. $PI$ cuts $\Omega$ and $\omega$ at $K, L$ respectively. Construct parallelogram $KFRE$. Prove that $A, R, L$ are collinear.
4 replies
Zavyk09
Yesterday at 12:50 PM
Zavyk09
4 hours ago
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shade from tub
QueenArwen   1
N 5 hours ago by mikestro
Source: 46th International Tournament of Towns, Senior O-Level P4, Spring 2025
There was a tub on the plane, with its upper base greater that the lower one. The tub was overturned. Prove that the area of its visible shade did decrease. (The tub is a frustum of a right circular cone: its bases are two discs in parallel planes, such that their centers lie on a line perpendicular to these planes. The visible shade is the total shade besides the shade under the tub. Consider the sun rays as parallel.)
1 reply
QueenArwen
Mar 11, 2025
mikestro
5 hours ago
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Inequality
Sappat   10
N 5 hours ago by iamnotgentle
Let $a,b,c$ be real numbers such that $a^2+b^2+c^2=1$. Prove that
$\frac{a^2}{1+2bc}+\frac{b^2}{1+2ca}+\frac{c^2}{1+2ab}\geq\frac{3}{5}$
10 replies
Sappat
Feb 7, 2018
iamnotgentle
5 hours ago
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ISI UGB 2025 P4
SomeonecoolLovesMaths   9
N 5 hours ago by nyacide
Source: ISI UGB 2025 P4
Let $S^1 = \{ z \in \mathbb{C} \mid |z| =1 \}$ be the unit circle in the complex plane. Let $f \colon S^1 \longrightarrow S^2$ be the map given by $f(z) = z^2$. We define $f^{(1)} \colon = f$ and $f^{(k+1)} \colon = f \circ f^{(k)}$ for $k \geq 1$. The smallest positive integer $n$ such that $f^{(n)}(z) = z$ is called the period of $z$. Determine the total number of points in $S^1$ of period $2025$.
(Hint : $2025 = 3^4 \times 5^2$)
9 replies
SomeonecoolLovesMaths
May 11, 2025
nyacide
5 hours ago
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J
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hard problem
Cobedangiu   8
N Apr 23, 2025 by IceyCold
Let $x,y,z>0$ and $xy+yz+zx=3$ : Prove that :
$\sum \ \frac{x}{y+z}\ge\sum \frac{1}{\sqrt{x+3}}$
8 replies
Cobedangiu
Apr 2, 2025
IceyCold
Apr 23, 2025
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hard problem
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Reveal topic tags
inequalities
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Cobedangiu
70 posts
Cobedangiu
#1 h  Apr 2, 2025, 6:11 PM
Y by
Let $x,y,z>0$ and $xy+yz+zx=3$ : Prove that :
$\sum \ \frac{x}{y+z}\ge\sum \frac{1}{\sqrt{x+3}}$
Z K Y
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Cobedangiu
70 posts
Cobedangiu
#4 h  Apr 3, 2025, 9:03 AM
Y by
...........
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arqady
30253 posts
arqady
#5 h  Apr 3, 2025, 10:37 AM
Y by
Cobedangiu wrote:
Let $x,y,z>0$ and $xy+yz+zx=3$ : Prove that :
$\sum \ \frac{x}{y+z}\ge\sum \frac{1}{\sqrt{x+3}}$
Use Jensen and $uvw$.
Z K Y
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Math-Problem-Solving
66 posts
Math-Problem-Solving
#6 h  Apr 22, 2025, 3:32 PM
Y by
arqady wrote:
Cobedangiu wrote:
Let $x,y,z>0$ and $xy+yz+zx=3$ : Prove that :
$\sum \ \frac{x}{y+z}\ge\sum \frac{1}{\sqrt{x+3}}$
Use Jensen and $uvw$.

What does this $uvw$ stand for?
This post has been edited 1 time. Last edited by Math-Problem-Solving, Apr 22, 2025, 3:33 PM
Reason: Err
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giangtruong13
146 posts
giangtruong13
#7 h  Apr 22, 2025, 3:39 PM
Y by
$x+y+z=u$,$xy+yz+zx=v$,$xyz=w$
This post has been edited 1 time. Last edited by giangtruong13, Apr 22, 2025, 3:39 PM
Z K Y
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IceyCold
209 posts
IceyCold
#8 h  Apr 22, 2025, 3:54 PM
Y by
giangtruong13 wrote:
$x+y+z=u$,$xy+yz+zx=v$,$xyz=w$

That's $pqr$,not $uvw$.

$uvw$ should be $a+b+c=3u , ab+bc+ca=3v^2 , abc=w^3.$
Z K Y
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Jackson0423
99 posts
Jackson0423
#9 h  Apr 22, 2025, 4:01 PM
Y by
pqr LMAO
I prefer pqr
This post has been edited 1 time. Last edited by Jackson0423, Apr 22, 2025, 4:01 PM
Reason: d
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arqady
30253 posts
arqady
#10 h  Apr 22, 2025, 7:48 PM
Y by
Math-Problem-Solving wrote:

What does this $uvw$ stand for?
$x+y+z=3u$, $xy+xz+yz=3v^2$ and $xyz=w^3$.
Z K Y
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IceyCold
209 posts
IceyCold
#11 h  Apr 23, 2025, 1:52 PM
Y by
Jackson0423 wrote:
$pqr$ LMAO
I prefer $pqr$

I also prefer $pqr$,but $uvw$ just seems stronger-
Why is this though?
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