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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
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0 replies
jlacosta
Apr 2, 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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Convex geometry
ILOVEMYFAMILY   3
N 27 minutes ago by ILOVEMYFAMILY
1) Find all closed convex sets with nonempty interior that have exactly one supporting hyperplane in the plane.

2) Find all closed convex sets with nonempty interior that have exactly two supporting hyperplane in the plane.

3 replies
ILOVEMYFAMILY
Apr 15, 2025
ILOVEMYFAMILY
27 minutes ago
J
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Apple sharing in Iran
mojyla222   0
33 minutes ago
Source: Iran 2025 second round p6
Ali is hosting a large party. Together with his $n-1$ friends, $n$ people are seated around a circular table in a fixed order. Ali places $n$ apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction).

Find all values of $n$ such that after some number of steps, the situation reaches a point where each person has exactly one apple.
0 replies
mojyla222
33 minutes ago
0 replies
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number of separated partitions for n+1 is equal the number of partitions for n
YLG_123   5
N 39 minutes ago by Victor23TT
Source: Brazilian Mathematical Olympiad 2024, Level 3, Problem 2
A partition of a set \( A \) is a family of non-empty subsets of \( A \), such that any two distinct subsets in the family are disjoint, and the union of all subsets equals \( A \). We say that a partition of a set of integers \( B \) is separated if each subset in the partition does not contain consecutive integers. Prove that, for every positive integer \( n \), the number of partitions of the set \( \{1, 2, \dots, n\} \) is equal to the number of separated partitions of the set \( \{1, 2, \dots, n+1\} \).

For example, \( \{\{1,3\}, \{2\}\} \) is a separated partition of the set \( \{1,2,3\} \). On the other hand, \( \{\{1,2\}, \{3\}\} \) is a partition of the same set, but it is not separated since \( \{1,2\} \) contains consecutive integers.
5 replies
YLG_123
Oct 12, 2024
Victor23TT
39 minutes ago
J
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Find the limit
Butterfly   0
an hour ago
$$\lim_{n \to \infty} \sum_{k=1}^n\left(\frac{1}{\sqrt{k^2+k}}-\ln\left(1+\frac{1}{k}\right)\right).$$
0 replies
Butterfly
an hour ago
0 replies
J
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Inspired by old results
sqing   5
N an hour ago by sqing
Source: Own
Let $ a,b>0. $ Prove that
$$\frac{(a+1)^2}{b}+\frac{(b+k)^2}{a} \geq4(k+1) $$Where $ k\geq 0. $
$$\frac{a^2}{b}+\frac{(b+1)^2}{a} \geq4$$
5 replies
sqing
Yesterday at 2:43 AM
sqing
an hour ago
J
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a+d=2^k and b+c=2^m for some integers k and m
ehsan2004   15
N 2 hours ago by SwordAxe
Source: IMO 1984, Day 2, Problem 6
Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.
15 replies
ehsan2004
Feb 12, 2005
SwordAxe
2 hours ago
J
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Inspired by Bet667
sqing   0
2 hours ago
Source: Own
Let $ x,y\ge 0 $ such that $k(x+y)=1+xy. $ Prove that$$x+k^2y+\frac{1}{x}+\frac{k^2}{y} \geq \frac{k^2(k+1)^2+(k-1)^2}{k}$$Where $ k\in N^+.$
Let $ x,y\ge 0 $ such that $2(x+y)=1+xy. $ Prove that$$x+4y+\frac{1}{x}+\frac{4}{y} \geq \frac{37}{2}$$
0 replies
sqing
2 hours ago
0 replies
J
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Is it always possible to color the points red or white?
orl   7
N 2 hours ago by SwordAxe
Source: 1986, Day 2, Problem 6
Given a finite set of points in the plane, each with integer coordinates, is it always possible to color the points red or white so that for any straight line $L$ parallel to one of the coordinate axes the difference (in absolute value) between the numbers of white and red points on $L$ is not greater than $1$?
7 replies
orl
Nov 11, 2005
SwordAxe
2 hours ago
J
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Inspired by old results
sqing   2
N 2 hours ago by sqing
Source: Own
Let $ a,b, c\geq 0 $ and $ a+b+c +a^2+b^2+c^2= 4$. Prove that
$$ (a^3+b^3)(b^3+c^3 )(c^3+a^3)\le 2$$
2 replies
sqing
Yesterday at 12:37 PM
sqing
2 hours ago
J
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3 var inquality
sqing   5
N 2 hours ago by sqing
Source: Own
Let $ a,b,c $ be reals such that $ a+b+c=0 $ and $ abc\geq \frac{1}{\sqrt{2}} . $ Prove that
$$ a^2+b^2+c^2\geq 3$$Let $ a,b,c $ be reals such that $ a+2b+c=0 $ and $ abc\geq \frac{1}{\sqrt{2}} . $ Prove that
$$ a^2+b^2+c^2\geq \frac{3}{ \sqrt[3]{2}}$$$$ a^2+2b^2+c^2\geq 2\sqrt[3]{4} $$
5 replies
sqing
Yesterday at 8:32 AM
sqing
2 hours ago
J
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Substitutions inequality?
giangtruong13   3
N 2 hours ago by giangtruong13
Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{b+c}{a} \geq 2\sqrt{3(ab+bc+ca)}$$
3 replies
giangtruong13
Friday at 2:07 PM
giangtruong13
2 hours ago
J
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functional equation on natural numbers ! CMO 2015 P1
aditya21   18
N 2 hours ago by NicoN9
Source: Canadian mathematical olympiad 2015
Let $\mathbb{N} = \{1, 2, 3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^2< f(n)f(f(n)) < n^2+n$ for every positive integer $n$.
18 replies
aditya21
Apr 24, 2015
NicoN9
2 hours ago
J
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fractional binomial limit sum
Levieee   1
N 5 hours ago by KAME06
this was given to me by a friend

$\lim_{n \to \infty} \sum_{k=0}^{n}{\frac{1}{\binom{n}{k}}}$

a nice solution using sandwich is
$\frac{1}{n} + \frac{1}{n} + 1 + \frac{n-3}{\binom{n}{2}} \ge \frac{1}{n} + \sum_{k=1}^{n-2}{\frac{1}{\binom{n}{k}}}+ \frac{1}{n} + 1 \ge \frac{1}{n} + + \frac{1}{n} + 1$

therefore $\lim_{n \to \infty} \sum_{k=0}^{n}{\frac{1}{\binom{n}{k}}}$ = $1$

ALSO ANOTHER SOLUTION WHICH I WAS THINKING OF WITHOUT SANDWICH BUT I CANT COMPLETE WAS TO USE THE GAMMA FUNCTION

we know

$B(x, y) = \int_0^1 t^{x - 1} (1 - t)^{y - 1} \, dt$

$B(x, y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x + y)}$

and $\Gamma(n) = (n-1)!$ for integers,

$\frac{1}{\binom{n}{k}}$ = $\frac{k! (n-k)!}{n!}$

therefore from the gamma function we get

$ (n+1) \int_{0}^{1} x^k (1-x)^{n-k} dx$ = $\frac{1}{\binom{n}{k}}$ = $\frac{k! (n-k)!}{n!}$
$\Rightarrow$ $\lim_{n \to \infty} (n+1) \int_{0}^{1} \sum_{k=0}^{n} x^k (1-x)^{n-k} dx$ $=\lim_{n \to \infty} \sum_{k=0}^{n}{\frac{1}{\binom{n}{k}}}$

somehow im supposed to show that

$\lim_{n \to \infty} (n+1) \int_{0}^{1} \sum_{k=0}^{n} x^k (1-x)^{n-k} dx$ $= 1$

all i could observe was if we do L'hopital (which i hate to do as much as you do)

i get $\frac{ \int_{0}^{1} \sum_{k=0}^{n} x^k (1-x)^{n-k} dx}{1/n+1}$

now since $x \in (0,1)$ , as $n \to \infty$ the $(1-x)^{n-k} \to 0$ which gets us the $\frac{0}{0}$ form therefore L'hopital came to my mind , which might be a completely wrong intuition, anyway what should i do to find that limit

:noo: :pilot:
1 reply
Levieee
Yesterday at 9:51 PM
KAME06
5 hours ago
J
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ap calculus bc
needcalculusasap45   1
N Yesterday at 9:12 PM by needcalculusasap45
So basically, I have the AP Calculus BC exam in less than a month, and I have only covered until Unit 6 or 7 of the cirriculum. I am self studying this course (no teacher) and have not had much time to study bc of 6 other APs. I need to finish 8, 9, and 10 in less than 2 weeks. What can I do ? I would appreciate any help or resources anyone could provide. Could I just learn everything from barrons and princeton? Also, I have not taken AP Calculus AB before.
1 reply
needcalculusasap45
Yesterday at 1:55 PM
needcalculusasap45
Yesterday at 9:12 PM
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J
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real analysis
ay19bme   1
N Apr 6, 2025 by alexheinis
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1 reply
ay19bme
Apr 6, 2025
alexheinis
Apr 6, 2025
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real analysis
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ay19bme
#1 h  Apr 6, 2025, 10:04 AM
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alexheinis
10547 posts
alexheinis
#2 h  Apr 6, 2025, 12:59 PM
Y by
The limit equals $\int_0^1 \{1/x\}dx $ which we will calculate in another way.
We have $\int_{1/(n+1)}^1 \{1/x\}dx=\sum_1^n \int_{1/(k+1)}^{1/k} (1/x-[1/x])dx=\int_{1/(n+1)}^1 dx/x- \sum_1^n {1\over {k+1}}=$
$\ln(n+1)-\sum_2^{n+1} 1/k=1-(\sum_1^{n+1} 1/k-\ln(n+1))\rightarrow 1-\gamma$.
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