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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
5 hours ago
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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
1 viewing
jlacosta
5 hours ago
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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Queue geo
vincentwant   4
N 4 minutes ago by vincentwant
Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $Y, Z$ be the feet of the altitudes from $B, C$ to $AC, AB$ respectively. Let $D$ be the midpoint of $BC$. Let $\omega_1$ be the circle with diameter $AD$. Let $Q\neq A$ be the intersection of $(ABC)$ and $\omega$. Let $H$ be the orthocenter of $ABC$. Let $K$ be the intersection of $AQ$ and $BC$. Let $l_1,l_2$ be the lines through $Q$ tangent to $\omega,(AYZ)$ respectively. Let $I$ be the intersection of $l_1$ and $KH$. Let $P$ be the intersection of $l_2$ and $YZ$. Let $l$ be the line through $I$ parallel to $HD$ and let $O'$ be the reflection of $O$ across $l$. Prove that $O'P$ is tangent to $(KPQ)$.
4 replies
+1 w
vincentwant
Wednesday at 3:54 PM
vincentwant
4 minutes ago
J
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Inequality with 4 variables
bel.jad5   3
N 10 minutes ago by sqing
Source: Own
Let $a$,$b$,$c$ $d$ positive real numbers. Prove that:
\[ \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a} \geq 4+\frac{8(a-d)^2}{(a+b+c+d)^2}\]
3 replies
+1 w
bel.jad5
Sep 5, 2018
sqing
10 minutes ago
J
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Do not try to bash on beautiful geometry
ItzsleepyXD   9
N 12 minutes ago by Captainscrubz
Source: Own , Mock Thailand Mathematic Olympiad P9
Let $ABC$be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$
9 replies
ItzsleepyXD
Wednesday at 9:30 AM
Captainscrubz
12 minutes ago
J
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A sequence containing every natural number exactly once
Pomegranat   1
N 24 minutes ago by YaoAOPS
Source: Own
Does there exist a sequence \( \{a_n\}_{n=1}^{\infty} \), which is a permutation of the natural numbers (that is, each natural number appears exactly once), such that for every \( n \in \mathbb{N} \), the sum of the first \( n \) terms is divisible by \( n \)?
1 reply
1 viewing
Pomegranat
27 minutes ago
YaoAOPS
24 minutes ago
J
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Trigonometry article for geometry
xytunghoanh   0
an hour ago
Does anyone have any articles on using trigonometry to prove geometry problems (Law of Sines, Ceva's Theorem in trigonometric form,..) that they can share with me?
Thanks!
0 replies
1 viewing
xytunghoanh
an hour ago
0 replies
J
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centroid lies outside of triangle (not clickbait)
Scilyse   1
N an hour ago by LoloChen
Source: 数之谜 January (CHN TST Mock) Problem 5
Let $P$ be a convex polygon with centroid $G$, and let $\mathcal P$ be the set of vertices of $P$. Let $\mathcal X$ be the set of triangles with vertices all in $\mathcal P$. We sort the elements $\triangle ABC$ of $\mathcal X$ into the following three types:
[list]
[*] (Type 1) $G$ lies in the strict interior of $\triangle ABC$; let $\mathcal A$ be the set of triangles of this type.
[*] (Type 2) $G$ lies in the strict exterior of $\triangle ABC$; let $\mathcal B$ be the set of triangles of this type.
[*] (Type 3) $G$ lies on the boundary of $\triangle ABC$.
[/list]
For any triangle $T$, denote by $S_T$ the area of $T$. Prove that \[\sum_{T \in \mathcal A} S_T \geq \sum_{T \in \mathcal B} S_T.\]
1 reply
1 viewing
Scilyse
Jan 26, 2025
LoloChen
an hour ago
J
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Function equation
LeDuonggg   1
N an hour ago by luutrongphuc
Find all functions $f: \mathbb{R^+} \rightarrow \mathbb{R^+}$ , such that for all $x,y>0$:
\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]
1 reply
LeDuonggg
Yesterday at 2:59 PM
luutrongphuc
an hour ago
J
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4 lines concurrent
Zavyk09   6
N an hour ago by hectorleo123
Source: Homework
Let $ABC$ be triangle with circumcenter $(O)$ and orthocenter $H$. $BH, CH$ intersect $(O)$ again at $K, L$ respectively. Lines through $H$ parallel to $AB, AC$ intersects $AC, AB$ at $E, F$ respectively. Point $D$ such that $HKDL$ is a parallelogram. Prove that lines $KE, LF$ and $AD$ are concurrent at a point on $OH$.
6 replies
Zavyk09
Apr 9, 2025
hectorleo123
an hour ago
J
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No More than √㏑x㏑㏑x Digits
EthanWYX2009   4
N 2 hours ago by tom-nowy
Source: 2024 April 谜之竞赛-3
Let $f(x)\in\mathbb Z[x]$ have positive integer leading coefficient. Show that there exists infinte positive integer $x,$ such that the number of digit that doesn'r equal to $9$ is no more than $\mathcal O(\sqrt{\ln x\ln\ln x}).$

Created by Chunji Wang, Zhenyu Dong
4 replies
EthanWYX2009
Mar 24, 2025
tom-nowy
2 hours ago
J
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Old hard problem
ItzsleepyXD   1
N 2 hours ago by ItzsleepyXD
Source: IDK
Let $ABC$ be a triangle and let $O$ be its circumcenter and $I$ its incenter.
Let $P$ be the radical center of its three mixtilinears and let $Q$ be the isogonal conjugate of $P$.
Let $G$ be the Gergonne point of the triangle $ABC$.
Prove that line $QG$ is parallel with line $OI$ .
1 reply
ItzsleepyXD
Apr 25, 2025
ItzsleepyXD
2 hours ago
J
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Existence of a solution of a diophantine equation
syk0526   5
N 2 hours ago by cursed_tangent1434
Source: North Korea Team Selection Test 2013 #6
Show that $ x^3 + x+ a^2 = y^2 $ has at least one pair of positive integer solution $ (x,y) $ for each positive integer $ a $.
5 replies
syk0526
May 17, 2014
cursed_tangent1434
2 hours ago
J
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Inequality with 3 variables
sqing   0
2 hours ago
Source: Own
Let $ a,b,c\geq 0 ,a^3b^3+b^3c^3+c^3a^3+2abc\geq 1 . $ Prove that$$a+b+c\geq 2 $$Let $ a,b,c\geq 0 ,a^3b^3+b^3c^3+c^3a^3+6abc\geq 9 . $ Prove that$$a+b+c\geq 2\sqrt 3 $$Let $ a,b,c\geq 0 ,a^3b+b^3c+c^3a+6abc\geq 9 . $ Prove that$$a+b+c\geq 3 $$Let $ a,b,c\geq 0 ,a^3b+b^3c+c^3a+3abc\geq 3 . $ Prove that$$a+b+c\geq \frac{4}{\sqrt 3} $$
0 replies
sqing
2 hours ago
0 replies
J
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Inequality with 3 variables and a special condition
Nuran2010   5
N 2 hours ago by sqing
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
For positive real numbers $a,b,c$ we have $3abc \geq ab+bc+ca$.
Prove that:

$\frac{1}{a^3+b^3+c}+\frac{1}{b^3+c^3+a}+\frac{1}{c^3+a^3+b} \leq \frac{3}{a+b+c}$.

Determine the equality case.
5 replies
Nuran2010
Apr 29, 2025
sqing
2 hours ago
J
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Chain of floors
Assassino9931   0
3 hours ago
Source: Vojtech Jarnik IMC 2025, Category I, P2
Determine all real numbers $x>1$ such that
\[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \]for any positive integer $n$.
0 replies
Assassino9931
3 hours ago
0 replies
J
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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
67 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
67 posts
Cobedangiu
#4 h  Apr 3, 2025, 9:03 AM
Y by
...........
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arqady
30228 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
139 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
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IceyCold
208 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
57 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
30228 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
208 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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