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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
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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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CSMGO P3: A problem on the infamous line XH
amar_04   12
N an hour ago by WLOGQED1729
Source: https://artofproblemsolving.com/community/c594864h2372843p19407517
Let $\triangle ABC$ be a scalene triangle with the orthocenter $H$. Let $B'$ be the reflection of $B$ over $AC$ and $C'$ be the reflection of $C$ over $AB$. Let the tangents to the circumcircle of $\triangle ABC$ at points $B$ and $C$ meet at a point $X$. Suppose that the lines $B'C'$ and $BC$ meet at a point $T$. Prove that $AT$ is perpendicular to $XH$.
12 replies
1 viewing
amar_04
Feb 16, 2021
WLOGQED1729
an hour ago
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Hard Function
johnlp1234   11
N an hour ago by GreekIdiot
f:R+--->R+:
f(x^3+f(y))=y+(f(x))^3
11 replies
johnlp1234
Jul 7, 2020
GreekIdiot
an hour ago
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Three mutually tangent circles
math154   8
N an hour ago by lakshya2009
Source: ELMO Shortlist 2011, G2
Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$, $\omega_1,\omega$ are internally tangent at $A$, and $\omega,\omega_2$ are internally tangent at $B$. Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$, respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$, prove that $\angle{O_1XP}=\angle{O_2XP}$.

David Yang.
8 replies
math154
Jul 3, 2012
lakshya2009
an hour ago
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Line AT passes through either S_1 or S_2
v_Enhance   89
N an hour ago by zuat.e
Source: USA December TST for 57th IMO 2016, Problem 2
Let $ABC$ be a scalene triangle with circumcircle $\Omega$, and suppose the incircle of $ABC$ touches $BC$ at $D$. The angle bisector of $\angle A$ meets $BC$ and $\Omega$ at $E$ and $F$. The circumcircle of $\triangle DEF$ intersects the $A$-excircle at $S_1$, $S_2$, and $\Omega$ at $T \neq F$. Prove that line $AT$ passes through either $S_1$ or $S_2$.

Proposed by Evan Chen
89 replies
v_Enhance
Dec 21, 2015
zuat.e
an hour ago
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Pertenacious Polynomial Problem
BadAtCompetitionMath21420   2
N an hour ago by Tetra_scheme
Let the polynomial $P(x) = x^3-x^2+px-q$ have real roots and real coefficients with $q>0$. What is the maximum value of $p+q$?

This is a problem I made for my math competition, and I wanted to see if someone would double-check my work (No Mike allowed):

solution
Is this solution good?
2 replies
1 viewing
BadAtCompetitionMath21420
Today at 3:13 AM
Tetra_scheme
an hour ago
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Easy geo
kooooo   3
N an hour ago by Blackbeam999
Source: own
In triangle $ABC$, let $O$ and $H$ be the circumcenter and orthocenter, respectively. Let $M$ and $N$ be the midpoints of $AC$ and $AB$, respectively, and let $D$ and $E$ be the feet of the perpendiculars from $B$ and $C$ to the opposite sides, respectively. Show that if $X$ is the intersection of $MN$ and $DE$, then $AX$ is perpendicular to $OH$.
3 replies
kooooo
Jul 31, 2024
Blackbeam999
an hour ago
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Interesting
imnotgoodatmathsorry   0
an hour ago
Source: Own.
Problem 1. Let $x,y,z >0$. Prove that:
$\frac{108(x^6+y^6)(y^6+z^6)(z^6+x^6)}{x^9y^9z^9} - (xy+yz+zx)^6 \le 135$
Problem 2. Let $a,b,c >0$. Prove that:
$(a+b+c)^4(ab+bc+ca) - 9\sum{\frac{a}{c}} \ge 54[(a+b)(b+c)(c+a)+abc-1]$
0 replies
imnotgoodatmathsorry
an hour ago
0 replies
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$n^{22}-1$ and $n^{40}-1$
v_Enhance   5
N an hour ago by Kempu33334
Source: OTIS Mock AIME 2024 #13
Let $S$ denote the sum of all integers $n$ such that $1 \leq n \leq 2024$ and exactly one of $n^{22}-1$ and $n^{40}-1$ is divisible by $2024$. Compute the remainder when $S$ is divided by $1000$.

Raymond Zhu

5 replies
v_Enhance
Jan 16, 2024
Kempu33334
an hour ago
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Annoying 2^x-5 = 11^y
Valentin Vornicu   38
N an hour ago by Kempu33334
Find all positive integer solutions to $2^x - 5 = 11^y$.

Comment (some ideas)
38 replies
Valentin Vornicu
Jan 14, 2006
Kempu33334
an hour ago
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Polish MO Finals 2014, Problem 5
j___d   14
N an hour ago by Kempu33334
Source: Polish MO Finals 2014
Find all pairs $(x,y)$ of positive integers that satisfy
$$2^x+17=y^4$$.
14 replies
j___d
Jul 27, 2016
Kempu33334
an hour ago
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IMO LongList 1985 CYP2 - System of Simultaneous Equations
Amir Hossein   15
N an hour ago by Kempu33334
Solve the system of simultaneous equations
\[\sqrt x - \frac 1y - 2w + 3z = 1,\]\[x + \frac{1}{y^2} - 4w^2 - 9z^2 = 3,\]\[x \sqrt x - \frac{1}{y^3} - 8w^3 + 27z^3 = -5,\]\[x^2 + \frac{1}{y^4} - 16w^4 - 81z^4 = 15.\]
15 replies
Amir Hossein
Sep 10, 2010
Kempu33334
an hour ago
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2022 MARBLE - Mock ARML I -8 \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=32
parmenides51   2
N an hour ago by Kempu33334
Let $a,b,c$ complex numbers with $ab+ +bc+ca = 61$ such that
$$\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{a+b}= 5$$$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=32.$$Find the value of $abc$.
2 replies
parmenides51
Jan 14, 2024
Kempu33334
an hour ago
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Inequalities
sqing   13
N 3 hours ago by sqing
Let $ a,b,c>0 , a+b+c +abc=4$. Prove that
$$ \frac {a}{a^2+2}+\frac {b}{b^2+2}+\frac {c}{c^2+2} \leq 1$$Let $ a,b,c>0 , ab+bc+ca+abc=4$. Prove that
$$ \frac {a}{a^2+2}+\frac {b}{b^2+2}+\frac {c}{c^2+2} \leq 1$$
13 replies
sqing
May 15, 2025
sqing
3 hours ago
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Geomettry ez
AnhIsGod   1
N 5 hours ago by Soupboy0
Let two circles (O) and (O') intersect at two points (one of which is called A). The common tangent CD (with C belonging to (O) and D belonging to (O')) lies on the same side as A with respect to the line OO', intersecting OO' at S. The line segment SA intersects circle (O) at E (different from A). Prove that EC is parallel to AD.
1 reply
AnhIsGod
5 hours ago
Soupboy0
5 hours ago
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An inequality
JK1603JK   3
N Apr 1, 2025 by lbh_qys
Let a,b,c\ge 0: ab+bc+ca>0 then prove \frac{4ab+5c^2}{a+b}+\frac{4bc+5a^2}{b+c}+\frac{4ca+5b^2}{c+a}\ge \frac{3}{2}\cdot\frac{(a+b+c)^3}{ab+bc+ca}.
3 replies
JK1603JK
Mar 31, 2025
lbh_qys
Apr 1, 2025
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An inequality
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Reveal topic tags
inequalities
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JK1603JK
53 posts
JK1603JK
#1 h  Mar 31, 2025, 7:11 AM
Y by
Let a,b,c\ge 0: ab+bc+ca>0 then prove \frac{4ab+5c^2}{a+b}+\frac{4bc+5a^2}{b+c}+\frac{4ca+5b^2}{c+a}\ge \frac{3}{2}\cdot\frac{(a+b+c)^3}{ab+bc+ca}.
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lbh_qys
583 posts
lbh_qys
#2 h  Mar 31, 2025, 9:27 AM
Y by
JK1603JK wrote:
Let $a,b,c\ge 0$: $ab+bc+ca>0$ then prove $\frac{4ab+5c^2}{a+b}+\frac{4bc+5a^2}{b+c}+\frac{4ca+5b^2}{c+a}\ge \frac{3}{2}\cdot\frac{(a+b+c)^3}{ab+bc+ca}.$

It appears that this inequality can be proven using the following generalization of the Iran 96 inequality

\[ \frac{15}{16} \cdot \frac{\sum a(a-b)(a-c)}{\prod (a+b)} \leq \left(\sum ab\right)\left(\sum \frac{1}{(a+b)^2}\right) -\frac 94\leq \frac{\sum a(a-b)(a-c)}{\prod (a+b)} \]
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JK1603JK
53 posts
JK1603JK
#3 h  Apr 1, 2025, 8:14 AM
Y by
Can you explain more detail?
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lbh_qys
583 posts
lbh_qys
#4 h  Apr 1, 2025, 9:22 AM
Y by
We have
\[ \frac{4ab+5c^2}{a+b}=\frac{4(a+c)(b+c)}{a+b}+\frac{c^2}{a+b}-4c. \]According to the above inequality, we have
\[ 4\Bigl(\sum ab\Bigr) \sum \frac{(a+c)(b+c)}{a+b}=4\Bigl(\sum ab\Bigr)\prod (a+b)\sum\frac{1}{(a+b)^2}\ge 9\prod (a+b)+\frac{15}{4}\sum a(a-b)(a-c)\ge 9\prod (a+b)+\frac{1}{2}\sum a(a-b)(a-c), \]and
\[ \Bigl(\sum ab\Bigr)\sum\frac{c^2}{a+b}=\sum c^3+abc\sum\frac{c}{a+b}\ge \sum c^3+\frac{3}{2}abc. \]Thus, it suffices to prove
\[ 9\prod (a+b)+\frac{1}{2}\sum a(a-b)(a-c)+\sum c^3+\frac{3}{2}abc\ge \frac{3}{2}(a+b+c)^3+4(ab+bc+ca)(a+b+c). \]In fact, this is an identity; hence, it is only necessary to prove the generalized form of the Iran 96 inequality as stated above.
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