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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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[*]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
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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Reflections and midpoints in triangle
TUAN2k8   2
N 2 minutes ago by Double07
Source: Own
Given an triangle $ABC$ and a line $\ell$ in the plane.Let $A_1,B_1,C_1$ be reflections of $A,B,C$ across the line $\ell$, respectively.Let $D,E,F$ be the midpoints of $B_1C_1,C_1A_1,A_1B_1$, respectively.Let $A_2,B_2,C_2$ be the reflections of $A,B,C$ across $D,E,F$, respectively.Prove that the points $A_2,B_2,C_2$ lie on a line parallel to $\ell$.
2 replies
TUAN2k8
5 hours ago
Double07
2 minutes ago
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Cool Functional Equation
Warideeb   0
11 minutes ago
Find all functions real to real such that
$f(xy+f(x))=xf(y)+f(x)$
for all reals $x,y$.
0 replies
Warideeb
11 minutes ago
0 replies
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Simple inequality
sqing   59
N 14 minutes ago by ForeverSnow
Source: Shortlist BMO 2018, A1
Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that:

$$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$
59 replies
sqing
May 3, 2019
ForeverSnow
14 minutes ago
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Inspired by 2025 Xinjiang
sqing   1
N 25 minutes ago by OGMATH
Source: Own
Let $ a,b >0 . $ Prove that
$$ \left(1+\frac {a} { b}\right)\left(6+\frac {b}{a}\right) \left(2+\frac {a}{b}+\frac {b}{ a}\right) \geq\frac {625}{ 12}$$$$ \left(1+\frac {a} { b}\right)\left(6+\frac {a}{b}\right) \left(2+\frac {a}{b}+\frac {b}{ a}\right) \geq29+6\sqrt 6$$$$ \left(1+\frac {a} { b}\right)\left(2+\frac {b}{ a}\right) \left(2+\frac {a}{b}+\frac {b}{ a}\right) \geq \frac{3(63+11\sqrt{33})}{16} $$$$ \left(1+\frac {a} { b}\right)\left(2+\frac {a}{ b}\right) \left(2+\frac {a}{b}+\frac {b}{ a}\right) \geq \frac{223+70\sqrt{10}}{27} $$
1 reply
sqing
40 minutes ago
OGMATH
25 minutes ago
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Integer-Valued FE comes again
lminsl   208
N 27 minutes ago by megahertz13
Source: IMO 2019 Problem 1
Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$Proposed by Liam Baker, South Africa
208 replies
lminsl
Jul 16, 2019
megahertz13
27 minutes ago
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2025 Beijing High School Mathematics Competition Q9
sqing   2
N 28 minutes ago by sqing
Source: China
Let $ a,b,c,d >0 $ and $ (a^2+b^2+c^2)(b^2+c^2+d^2)=36. $ Find the maximum value of $a^2d.$
2 replies
sqing
2 hours ago
sqing
28 minutes ago
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2025 Guangdong High School Mathematics Competition Q14
sqing   2
N 28 minutes ago by sqing
Source: China
Let $ x_1, x_2, x_3, x_4, x_5\geq 0 $ and $ x^2_1+x^2_2+x^2_3+ x^2_4+ x^2_5=4. $ Find the maximum value of
$\sum_{i=1}^5 \frac{1}{x_i+1} \sum_{i=1}^5 x_i .$
2 replies
sqing
2 hours ago
sqing
28 minutes ago
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JBMO Shortlist 2023 G7
Orestis_Lignos   6
N 36 minutes ago by MR.1
Source: JBMO Shortlist 2023, G7
Let $D$ and $E$ be arbitrary points on the sides $BC$ and $AC$ of triangle $ABC$, respectively. The circumcircle of $\triangle ADC$ meets for the second time the circumcircle of $\triangle BCE$ at point $F$. Line $FE$ meets line $AD$ at point $G$, while line $FD$ meets line $BE$ at point $H$. Prove that lines $CF, AH$ and $BG$ pass through the same point.
6 replies
Orestis_Lignos
Jun 28, 2024
MR.1
36 minutes ago
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Inspired by 2025 Xinjiang
sqing   0
an hour ago
Source: Own
Let $ a,b,c >0 . $ Prove that
$$ \left(1+\frac {a} { b}\right)\left(2+\frac {a}{ b}+\frac {b}{ c}\right) \left(1+\frac {a}{b}+\frac {b}{ c}+\frac {c}{ a}\right) \geq 12+8\sqrt 2 $$
0 replies
sqing
an hour ago
0 replies
J
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2025 Xinjiang High School Mathematics Competition Q11
sqing   1
N an hour ago by sqing
Source: China
Let $ a,b,c >0 . $ Prove that
$$ \left(1+\frac {a} { b}\right)\left(1+\frac {a}{ b}+\frac {b}{ c}\right) \left(1+\frac {a}{b}+\frac {b}{ c}+\frac {c}{ a}\right) \geq 16 $$
1 reply
sqing
2 hours ago
sqing
an hour ago
J
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Nice "if and only if" function problem
ICE_CNME_4   6
N an hour ago by BBNoDollar
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[ f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0. \](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )

Please do it at 9th grade level. Thank you!
6 replies
ICE_CNME_4
Yesterday at 7:23 PM
BBNoDollar
an hour ago
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D1037 : The super irreductibilty
Dattier   0
an hour ago
Source: les dattes à Dattier
Let $P \in \mathbb Q[x]$, $P$ is super irreductible if $\forall Q \in \mathbb [x], \deg(Q)>0, P(Q(x))$ is irreductible.

Are there some $P \in \mathbb Q[x]$ super irreductible?
0 replies
Dattier
an hour ago
0 replies
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Inspired by 2025 Beijing
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b,c,d >0 $ and $ (a^2+b^2+c^2)(b^2+c^2+d^2)=36. $ Prove that
$$ab^2c^2d \leq 8$$$$a^2bcd^2 \leq 16$$$$ ab^3c^3d \leq \frac{2187}{128}$$$$ a^3bcd^3 \leq \frac{2187}{32}$$
1 reply
sqing
2 hours ago
sqing
an hour ago
J
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Arrange marbles
FunGuy1   5
N 2 hours ago by FunGuy1
Source: Own?
Anna has $200$ marbles in $25$ colors such that there are exactly $8$ marbles of each color. She wants to arrange them on $50$ shelves, $4$ marbles on each shelf such that for any $2$ colors there is a shelf that has marbles of those colors.
Can Anna achieve her goal?
5 replies
FunGuy1
Today at 12:23 PM
FunGuy1
2 hours ago
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Geometry inmo problem
Enes040612   2
N Apr 25, 2025 by Red_gear5246
Source: INMO 2008
Can anyone help with this problem? the book also starts the solution by saying IA1=IB1=IC1=2r which I'm not understanding why. The problem is at the attachments.
2 replies
Enes040612
Feb 13, 2025
Red_gear5246
Apr 25, 2025
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Geometry inmo problem
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Reveal topic tags
geometry
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G H BBookmark kLocked kLocked NReply
Source: INMO 2008
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Enes040612
4 posts
Enes040612
#1 h  Feb 13, 2025, 8:10 AM
Y by
Can anyone help with this problem? the book also starts the solution by saying IA1=IB1=IC1=2r which I'm not understanding why. The problem is at the attachments.
Attachments:
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CHESSR1DER
69 posts
CHESSR1DER
#2 h  Feb 13, 2025, 9:20 AM
Y by
Let $IA1 \cap BC = H$. Then since $IH = r$ and $IH = A1H$, then $IA1= 2r$. Other are similar.
This post has been edited 2 times. Last edited by CHESSR1DER, Feb 13, 2025, 9:21 AM
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Red_gear5246
2 posts
Red_gear5246
#3 h  Apr 25, 2025, 1:29 PM
Y by
@Enes040612 which book are u solving?
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