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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 11:16 PM
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
Yesterday at 11:16 PM
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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Nice one
imnotgoodatmathsorry   0
13 minutes ago
Source: Own
With $x,y,z >0$.Prove that: $\frac{xy}{4y+4z+x} + \frac{yz}{4z+4x+y} +\frac{zx}{4x+4y+z} \le \frac{x+y+z}{9}$
0 replies
2 viewing
imnotgoodatmathsorry
13 minutes ago
0 replies
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Geometric inequality in quadrilateral
BBNoDollar   1
N 13 minutes ago by nabodorbuco2
Source: Romanian Mathematical Gazette 2025
Let ABCD be a convex quadrilateral with angles BAD and BCD obtuse, and let the points E, F ∈ BD, such that AE ⊥ BD and CF ⊥ BD.
Prove that 1/(AE*CF) ≥ 1/(AB*BC) + 1/(AD*CD) .
1 reply
BBNoDollar
2 hours ago
nabodorbuco2
13 minutes ago
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How many numbers ?
brokendiamond   0
14 minutes ago
How many 9-digit numbers can be formed using digits 1 to 9, with the condition that no even digits are adjacent to each other ?
0 replies
brokendiamond
14 minutes ago
0 replies
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-2 belongs to S
WakeUp   4
N 18 minutes ago by jasperE3
Source: Baltic Way 1996 Q12
Let $S$ be a set of integers containing the numbers $0$ and $1996$. Suppose further that any integer root of any non-zero polynomial with coefficients in $S$ also belongs to $S$. Prove that $-2$ belongs to $S$.
4 replies
WakeUp
Mar 19, 2011
jasperE3
18 minutes ago
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IMC 1994 D1 P2
j___d   4
N 19 minutes ago by Rohit-2006
Let $f\in C^1(a,b)$, $\lim_{x\to a^+}f(x)=\infty$, $\lim_{x\to b^-}f(x)=-\infty$ and $f'(x)+f^2(x)\geq -1$ for $x\in (a,b)$. Prove that $b-a\geq\pi$ and give an example where $b-a=\pi$.
4 replies
j___d
Mar 6, 2017
Rohit-2006
19 minutes ago
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ISI 2019 : Problem #2
integrated_JRC   39
N 20 minutes ago by Levieee
Source: I.S.I. 2019
Let $f:(0,\infty)\to\mathbb{R}$ be defined by $$f(x)=\lim_{n\to\infty}\cos^n\bigg(\frac{1}{n^x}\bigg)$$(a) Show that $f$ has exactly one point of discontinuity.
(b) Evaluate $f$ at its point of discontinuity.
39 replies
+2 w
integrated_JRC
May 5, 2019
Levieee
20 minutes ago
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Inspired by Kunihiko_Chikaya
sqing   0
28 minutes ago
Source: Own
Let $a,\ b,\ c$ be real numbers such that $ a+b+c=1 $ and $ a^2+b^2+c^2=23 . $ Prove that
$$ -47\leq a^3+b^3-3c\leq 78$$$$\frac{509-278\sqrt{34}}{27}\leq a^3+b^3-3abc\leq \frac{509+278\sqrt{34}}{27}$$
0 replies
sqing
28 minutes ago
0 replies
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analysis
Hello_Kitty   0
34 minutes ago
Prove or disprove:
1) If $u_n\geq 0$ such $\sum u_n $ converges then there is some $w_n\geq 0$ such that
$w_n\longrightarrow \infty $ and $\sum u_nw_n$ converges.
2) If $u_n\geq 0$ such $\sum u_n $ converges then there is some $w_n\geq 0$ such that
$\forall a\in\mathbb{R}, \; n^aw_n\longrightarrow \infty $ and $\sum u_nw_n$ converges .
0 replies
Hello_Kitty
34 minutes ago
0 replies
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Distributing coins in a circle
quacksaysduck   2
N 36 minutes ago by genius_007
Source: JOM 2025 Mock 1 P4
There are $n$ people arranged in a circle, and $n^{n^n}$ coins are distributed among them, where each person has at least $n^n$ coins. Each person is then assigned a random index number in $\{1,2,...n\}$ such that no two people have the same number. Then every minute, if $i$ is the number of minutes passed, the person with index number congruent to $i$ mod $n$ will give a coin to the person on his left or right. After some time, everyone has the same number of coins.

For what $n$ is this always possible, regardless of the original distribution of coins and index numbers?

(Proposed by Ho Janson)
2 replies
quacksaysduck
Jan 26, 2025
genius_007
36 minutes ago
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Familiar cyclic quad config
Rijul saini   11
N 39 minutes ago by ihategeo_1969
Source: India IMOTC Practice Test 1 Problem 2
Let $ABCD$ be a convex cyclic quadrilateral with circumcircle $\omega$. Let $BA$ produced beyond $A$ meet $CD$ produced beyond $D$, at $L$. Let $\ell$ be a line through $L$ meeting $AD$ and $BC$ at $M$ and $N$ respectively, so that $M,D$ (respectively $N,C$) are on opposite sides of $A$ (resp. $B$). Suppose $K$ and $J$ are points on the arc $AB$ of $\omega$ not containing $C,D$ so that $MK, NJ$ are tangent to $\omega$. Prove that $K,J,L$ are collinear.

Proposed by Rijul Saini
11 replies
Rijul saini
May 31, 2024
ihategeo_1969
39 minutes ago
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Inspired by Nice inequality
sqing   1
N 41 minutes ago by sqing
Source: Own
Let $ a,b,c >0 $. Show that
$$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\right)^2 \geq \frac{16}{k+3}\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}+k\right)$$Where $ k\geq 1.$
$$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\right)^2 \geq \frac{b}{a}+\frac{c}{b}+\frac{a}{c}+13$$$$\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+1\right)^2 \geq \frac{16}{5}\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}+2\right)$$
1 reply
sqing
5 hours ago
sqing
41 minutes ago
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Hard inequality
ys33   5
N an hour ago by sqing
Let $a, b, c, d>0$. Prove that
$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$.
5 replies
ys33
5 hours ago
sqing
an hour ago
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Putnam 2017 A2
Kent Merryfield   28
N an hour ago by kamatadu
Let $Q_0(x)=1$, $Q_1(x)=x,$ and
\[Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}\]for all $n\ge 2.$ Show that, whenever $n$ is a positive integer, $Q_n(x)$ is equal to a polynomial with integer coefficients.
28 replies
Kent Merryfield
Dec 3, 2017
kamatadu
an hour ago
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China Northern MO 2009 p4 CNMO
parkjungmin   1
N an hour ago by exoticc
Source: China Northern MO 2009 p4 CNMO
China Northern MO 2009 p4 CNMO

The problem is too difficult.
Is there anyone who can help me?
1 reply
parkjungmin
Apr 30, 2025
exoticc
an hour ago
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real analysis
ay19bme   4
N Apr 8, 2025 by ay19bme
.............
4 replies
ay19bme
Apr 7, 2025
ay19bme
Apr 8, 2025
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real analysis
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Reveal topic tags
real analysis
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ay19bme
277 posts
ay19bme
#1 h  Apr 7, 2025, 7:11 PM
Y by
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alexheinis
10574 posts
alexheinis
#2 h  Apr 7, 2025, 7:38 PM
Y by
With $1/k=a$ we can rewrite as $\sum_2^\infty {1\over {n(\ln n)^a}}$.
This converges iff $\int_p^\infty {{dx}\over {x(\ln x)^a}}=\int_q^\infty {{dt}\over {t^a}}$ converges.
This is true iff $a>1$ hence iff $k<1$.

@below: we have $n^{a\ln \ln n/\ln n}=(e^{\ln n})^{a\ln \ln n/\ln n}=e^{a\ln \ln n}=(e^{\ln \ln n})^a=(\ln n)^a$.
This post has been edited 1 time. Last edited by alexheinis, Apr 7, 2025, 8:16 PM
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ay19bme
277 posts
ay19bme
#3 h  Apr 7, 2025, 7:42 PM
Y by
Can you please explain a little how the given sum turns out to be equal with your sum.
Thanks
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Svenskerhaor
9 posts
Svenskerhaor
#4 h  Apr 7, 2025, 9:48 PM
Y by
ay19bme wrote:
Can you please explain a little how the given sum turns out to be equal with your sum.
Thanks

\begin{align*} n^{1+(ln(ln(n))/ln(n))}=nn^{ln(ln(n))/ln(n)}=n(e^{ln n})^{ln(ln(n))/ln(n)}=ne^{ln(ln(n))}=n(ln(n)) \end{align*}
This post has been edited 1 time. Last edited by Svenskerhaor, Apr 7, 2025, 9:50 PM
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ay19bme
277 posts
ay19bme
#5 h  Apr 8, 2025, 1:55 PM
Y by
what about the lower bound of $k$
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