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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
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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Sociable set of people
jgnr   23
N 5 minutes 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
+1 w
jgnr
Mar 3, 2012
quantam13
5 minutes ago
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diophantine equation
m4thbl3nd3r   0
8 minutes ago
Find all positive integers $n,k$ such that $$5^{2n+1}-5^n+1=k^2$$
0 replies
m4thbl3nd3r
8 minutes ago
0 replies
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A geometry problem
Lttgeometry   1
N 23 minutes 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
23 minutes ago
J
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interesting functional equation
tabel   0
24 minutes ago
Source: random romanian contest
Determine all functions \( f : (0, \infty) \to (0, \infty) \) that satisfy the functional equation:
\[ f(f(x)(1 + y)) = f(x) + f(xy), \quad \forall x, y > 0. \]
0 replies
tabel
24 minutes ago
0 replies
J
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Inequality em981
oldbeginner   15
N an hour ago by sqing
Source: Own
Let $a, b, c>0, a+b+c=3$. Prove that
\[\sqrt{a+\frac{9}{b+2c}}+\sqrt{b+\frac{9}{c+2a}}+\sqrt{c+\frac{9}{a+2b}}+\frac{2(ab+bc+ca)}{9}\ge\frac{20}{3}\]
15 replies
oldbeginner
Sep 22, 2016
sqing
an hour ago
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Functional equation
shobber   19
N 2 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
2 hours ago
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Prove the inequality
Butterfly   0
2 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
2 hours ago
0 replies
J
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Functional equation
shactal   1
N 2 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
4 hours ago
ariopro1387
2 hours ago
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interesting diophantiic fe in natural numbers
skellyrah   5
N 2 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
2 hours ago
J
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Non-linear Recursive Sequence
amogususususus   3
N 3 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
3 hours ago
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Inspired by 2025 Beijing
sqing   6
N 3 hours 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
$$ab^2c^2d \leq 8$$$$a^2bcd^2 \leq 16$$$$ ab^3c^3d \leq \frac{2187}{128}$$$$ a^3bcd^3 \leq \frac{2187}{32}$$
6 replies
sqing
Yesterday at 4:56 PM
sqing
3 hours ago
J
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Serbian selection contest for the IMO 2025 - P4
OgnjenTesic   2
N 3 hours ago by sqing-inequality-BUST
Source: Serbian selection contest for the IMO 2025
For a permutation $\pi$ of the set $A = \{1, 2, \ldots, 2025\}$, define its colorfulness as the greatest natural number $k$ such that:
- For all $1 \le i, j \le 2025$, $i \ne j$, if $|i - j| < k$, then $|\pi(i) - \pi(j)| \ge k$.
What is the maximum possible colorfulness of a permutation of the set $A$? Determine how many such permutations have maximal colorfulness.

Proposed by Pavle Martinović
2 replies
OgnjenTesic
May 22, 2025
sqing-inequality-BUST
3 hours ago
J
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Nice "if and only if" function problem
ICE_CNME_4   14
N 3 hours ago by wh0nix
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!
14 replies
ICE_CNME_4
Friday at 7:23 PM
wh0nix
3 hours ago
J
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2-var inequality
sqing   1
N 3 hours ago by sqing
Source: Own
Let $ a,b> 0 , ab(a+b+1) =3.$ Prove that$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{24}{(a+b)^2} \geq 8$$$$ \frac{a}{b^2}+\frac{b}{a^2}+\frac{49}{(a+ b)^2} \geq \frac{57}{4}$$Let $ a,b> 0 , (a+b)(ab+1) =4.$ Prove that$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{40}{(a+b)^2} \geq 12$$$$\frac{a}{b^2}+\frac{b}{a^2}+\frac{76}{(a+ b)^2} \geq 21$$
1 reply
sqing
4 hours ago
sqing
3 hours ago
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J
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Find the value
sqing   9
N Apr 20, 2025 by sqing
Source: 2025 Tsinghua University
Let $A= \lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right) . $ Find the value of $[100A] .$
9 replies
sqing
Apr 20, 2025
sqing
Apr 20, 2025
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Find the value
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Reveal topic tags
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value
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Source: 2025 Tsinghua University
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sqing
42397 posts
sqing
#1 h  Apr 20, 2025, 9:57 AM
Y by
Let $A= \lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right) . $ Find the value of $[100A] .$
This post has been edited 2 times. Last edited by sqing, Apr 20, 2025, 1:20 PM
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pco
23515 posts
pco
#3 h  Apr 20, 2025, 12:41 PM
Y by
sqing wrote:
Let $A=\lim_{n\to\infty}(\frac{\pi}{4}+\frac{1}{n})^n . $ Find the value of $[100A] .$
$A=0$ and so $\boxed{\lfloor 100A\rfloor=0}$
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Quantum-Phantom
276 posts
Quantum-Phantom
#4 h  Apr 20, 2025, 1:11 PM
Y by
It should be
\[A=\lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right).\]
Z K Y
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sqing
42397 posts
sqing
#5 h  Apr 20, 2025, 1:19 PM
Y by
Thank you very much.
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sqing
42397 posts
sqing
#6 h  Apr 20, 2025, 1:20 PM
Y by
Sorry.Thank pco.
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SunnyEvan
130 posts
SunnyEvan
#7 h  Apr 20, 2025, 1:24 PM
Y by
Quantum-Phantom wrote:
It should be
\[A=\lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right).\]

$$ A=\lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right) =e^{\lim_{n\to\infty}n ln\tan(\frac{\pi}{4}+\frac{1}{n})} $$let $t=\frac{1}{n}$
$$ e^{\lim_{n\to\infty}n ln\tan(\frac{\pi}{4}+\frac{1}{n})} = e^{\lim_{t\to 0} \frac{ln\tan(\frac{\pi}{4}+t)}{t}} = e^{\lim_{t\to 0} \frac{1}{tan(\frac{\pi}{4}+t)}(sec(\frac{\pi}{4}+t))^2}= e^{\lim_{t\to 0} \frac{1}{sin(\frac{\pi}{4}+t)cos(\frac{\pi}{4}+t)}}=e^2$$$[100A] =738$
This post has been edited 3 times. Last edited by SunnyEvan, Apr 20, 2025, 1:53 PM
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sqing
42397 posts
sqing
#9 h  Apr 20, 2025, 1:48 PM
Y by
Prove that $$\lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right) = e^2$$
Z K Y
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sqing
42397 posts
sqing
#10 h  Apr 20, 2025, 1:57 PM
Y by
sqing wrote:
Prove that $$\lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right) = e^2$$
https://artofproblemsolving.com/community/c7h1643781p10364775
https://math.stackexchange.com/questions/1750591/limit-of-tan-function?noredirect=1
*
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SunnyEvan
130 posts
SunnyEvan
#11 h  Apr 20, 2025, 2:00 PM
Y by
sqing wrote:
sqing wrote:
Prove that $$\lim_{n\to\infty}\tan^n\left(\frac{\pi}{4}+\frac{1}{n}\right) = e^2$$
https://artofproblemsolving.com/community/c7h1643781p10364775
https://math.stackexchange.com/questions/1750591/limit-of-tan-function?noredirect=1
*

Thank you very much !@Sqing . :-D Hospital always help with lots of problems .
This post has been edited 1 time. Last edited by SunnyEvan, Apr 20, 2025, 2:02 PM
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sqing
42397 posts
sqing
#12 h  Apr 20, 2025, 2:56 PM
Y by
Thanks.
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