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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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Inequality em981
oldbeginner   16
N 10 minutes ago by mihaig
Source: Own
Let $a, b, c>0, a+b+c=3$. Prove that
\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]
16 replies
oldbeginner
Sep 22, 2016
mihaig
10 minutes ago
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Nice FE over R+
doanquangdang   4
N 11 minutes ago by jasperE3
Source: collect
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that
\[x+f(yf(x)+1)=xf(x+y)+yf(yf(x))\]for all $x,y>0.$
4 replies
doanquangdang
Jul 19, 2022
jasperE3
11 minutes ago
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right triangle, midpoints, two circles, find angle
star-1ord   0
14 minutes ago
Source: Estonia Final Round 2025 8-3
In the right triangle $ABC$, $M$ is the midpoint of the hypotenuse $AB$. Point $D$ is chosen on the leg $BC$ so that the line segment $DM$ meets $(ACD)$ again at $K$ ($K\neq D$). Let $L$ be the reflection of $K$ in $M$. The circles $(ACD)$ and $(BCL)$ meet again at $N$ ($N\neq C$). Find the measure of $\angle KNL$.
0 replies
star-1ord
14 minutes ago
0 replies
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interesting functional equation
tabel   3
N 16 minutes ago by waterbottle432
Source: random romanian contest
Determine all functions \( f : (0, \infty) \to (0, \infty) \) that satisfy the functional equation:
\[ f(f(x)(1 + y)) = f(x) + f(xy), \quad \forall x, y > 0. \]
3 replies
tabel
an hour ago
waterbottle432
16 minutes ago
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Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   4
N 32 minutes ago by GreenTea2593
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
4 replies
OgnjenTesic
May 22, 2025
GreenTea2593
32 minutes ago
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pairs (m, n) such that a fractional expression is an integer
cielblue   2
N an hour ago by cielblue
Find all pairs $(m,\ n)$ of positive integers such that $\frac{m^3-mn+1}{m^2+mn+2}$ is an integer.
2 replies
cielblue
Yesterday at 8:38 PM
cielblue
an hour ago
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Sociable set of people
jgnr   23
N an hour ago by quantam13
Source: RMM 2012 day 1 problem 1
Given a finite number of boys and girls, a sociable set of boys is a set of boys such that every girl knows at least one boy in that set; and a sociable set of girls is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.)

(Poland) Marek Cygan
23 replies
jgnr
Mar 3, 2012
quantam13
an hour ago
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diophantine equation
m4thbl3nd3r   0
an hour ago
Find all positive integers $n,k$ such that $$5^{2n+1}-5^n+1=k^2$$
0 replies
m4thbl3nd3r
an hour ago
0 replies
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A geometry problem
Lttgeometry   1
N an hour ago by Funcshun840
Triangle $ABC$ has two isogonal conjugate points $P$ and $Q$. The circle $(BPC)$ intersects circle $(AP)$ at $R \neq P$, and the circle $(BQC)$ intersects circle $(AQ)$ at $S\neq Q$. Prove that $R$ and $S$ are isogonal conjugates in triangle $ABC$.
Note: Circle $(AP)$ is the circle with diameter $AP$, Circle $(AQ)$ is the circle with diameter $AQ$.
1 reply
Lttgeometry
Today at 4:03 AM
Funcshun840
an hour ago
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Functional equation
shobber   19
N 3 hours ago by Unique_solver
Source: Canada 2002
Let $\mathbb N = \{0,1,2,\ldots\}$. Determine all functions $f: \mathbb N \to \mathbb N$ such that
\[ xf(y) + yf(x) = (x+y) f(x^2+y^2) \]
for all $x$ and $y$ in $\mathbb N$.
19 replies
shobber
Mar 5, 2006
Unique_solver
3 hours ago
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Prove the inequality
Butterfly   0
3 hours ago
Let $a,b,c$ be real numbers such that $a+b+c=3$. Prove $$a^3b+b^3c+c^3a\le \frac{9}{32}(63+5\sqrt{105}).$$
0 replies
Butterfly
3 hours ago
0 replies
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Functional equation
shactal   1
N 3 hours ago by ariopro1387
Let $f:\mathbb R\to \mathbb R$ a function satifying $$f(x+2xy) = f(x) + 2f(xy)$$for all $x,y\in \mathbb R$.
If $f(1991)=a$, then what is $f(1992)$, the answer is in terms of $a$.
1 reply
shactal
5 hours ago
ariopro1387
3 hours ago
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interesting diophantiic fe in natural numbers
skellyrah   5
N 3 hours ago by skellyrah
Find all functions \( f : \mathbb{N} \to \mathbb{N} \) such that for all \( m, n \in \mathbb{N} \),
\[ mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!). \]
5 replies
skellyrah
Yesterday at 8:01 AM
skellyrah
3 hours ago
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Non-linear Recursive Sequence
amogususususus   3
N 4 hours ago by SunnyEvan
Given $a_1=1$ and the recursive relation
$$a_{i+1}=a_i+\frac{1}{a_i}$$for all natural number $i$. Find the general form of $a_n$.

Is there any way to solve this problem and similar ones?
3 replies
amogususususus
Jan 24, 2025
SunnyEvan
4 hours ago
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Non-negative real variables inequality
KhuongTrang   2
N Apr 29, 2025 by NguyenVanHoa29
Source: own
Problem. Let $a,b,c\ge 0: ab+bc+ca>0.$ Prove that$$\color{blue}{\frac{\left(2ab+ca+cb\right)^{2}}{a^{2}+4ab+b^{2}}+\frac{\left(2bc+ab+ac\right)^{2}}{b^{2}+4bc+c^{2}}+\frac{\left(2ca+bc+ba\right)^{2}}{c^{2}+4ca+a^{2}}\ge \frac{8(ab+bc+ca)}{3}.}$$
2 replies
KhuongTrang
Apr 24, 2025
NguyenVanHoa29
Apr 29, 2025
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Non-negative real variables inequality
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Reveal topic tags
inequalities
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Source: own
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KhuongTrang
731 posts
KhuongTrang
#1 h  Apr 24, 2025, 2:52 PM • 4 Y
Y by math90, NguyenVanHoa29, Zuyong, TNKT
Problem. Let $a,b,c\ge 0: ab+bc+ca>0.$ Prove that$$\color{blue}{\frac{\left(2ab+ca+cb\right)^{2}}{a^{2}+4ab+b^{2}}+\frac{\left(2bc+ab+ac\right)^{2}}{b^{2}+4bc+c^{2}}+\frac{\left(2ca+bc+ba\right)^{2}}{c^{2}+4ca+a^{2}}\ge \frac{8(ab+bc+ca)}{3}.}$$
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Quantum-Phantom
276 posts
Quantum-Phantom
#2 h  Apr 25, 2025, 3:31 AM
Y by
Are there any easier methods?

After multiplying both sides by \(\prod\limits_{\rm cyc}\left(a^2+4ab+b^2\right)\), we need to show that
\[\frac13\sum_{\rm cyc}a^2b^2(a+3b)(3a+b)(a-b)^2+abc\cdot f(a,b,c)\ge0,\]where $f(a,b,c)$ is a fifth degree polynomial:
\[\sum_{\rm cyc}\left(2a^5+\frac83a^4b+\frac83ab^4+\frac{22}3a^2b^2c-\frac{14}3a^3b^2-\frac{14}3a^2b^3-\frac{16}3a^2b^2c\right).\]By the $uvw$ method, it is not hard to show that $f(a,b,c)\ge0$ is true.

Click to reveal hidden text
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NguyenVanHoa29
9 posts
NguyenVanHoa29
#3 h  Apr 29, 2025, 9:51 AM • 1 Y
Y by arqady
I think it is a concave function according to w^3 and the rest is easy checking.
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