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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

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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
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
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
Convergence of complex sequence
Rohit-2006   2
N 25 minutes ago by c00lb0y
Suppose $z_1, z_2,\cdots,z_k$ are complex numbers with absolute value $1$. For $n=1,2,\cdots$ define $w_n=z_1^n+z_2^n+\cdots+z_k^n$. Given that the sequence $(w_n)_{n\geq1}$ converges. Show that,
$$z_1=z_2=\cdots=z_k=1$$.
2 replies
Rohit-2006
Saturday at 7:56 PM
c00lb0y
25 minutes ago
USAMO solutions
ABCD1728   1
N an hour ago by ohiorizzler1434
Do USAMO USATSTST and USATST have OFFICIAL solutions? Thanks!
1 reply
ABCD1728
2 hours ago
ohiorizzler1434
an hour ago
D1035 : Super TVI 2
Dattier   1
N an hour ago by alexheinis
Source: les dattes à Dattier
Let $f \in C([0,1])$. Is it true that $\exists a \in \left[0;\dfrac 13\right] \cup  \left[\dfrac 23; 1 \right]  , |f(a)| \leq 4 |f(1-a)|$ ?
1 reply
Dattier
Yesterday at 10:54 PM
alexheinis
an hour ago
D1034 : Super TVI
Dattier   2
N an hour ago by alexheinis
Source: les dattes à Dattier
Let $f \in C([0,1])$. Is it true that $\exists a \in \left[0;\dfrac 12\right], |f(a)| \leq 8 |f(1-a)|$ ?
2 replies
Dattier
Yesterday at 10:21 PM
alexheinis
an hour ago
Calculus
youochange   3
N 2 hours ago by c00lb0y
Find the area enclosed by the curves $e^x,e^{-x},x^2+y^2=1$

3 replies
youochange
May 10, 2025
c00lb0y
2 hours ago
Distance between any two points is irrational
orl   21
N 2 hours ago by cursed_tangent1434
Source: IMO 1987, Day 2, Problem 5
Let $n\ge3$ be an integer. Prove that there is a set of $n$ points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area.
21 replies
orl
Nov 11, 2005
cursed_tangent1434
2 hours ago
Symmetric inequality
mrrobotbcmc   5
N 2 hours ago by youochange
Let a,b,c,d belong to positive real numbers such that a+b+c+d=1. Prove that a^3/(b+c)+b^3/(c+d)+c^3/(d+a)+d^3/(a+b)>=1/8
5 replies
mrrobotbcmc
4 hours ago
youochange
2 hours ago
Two perpendiculars
jayme   0
2 hours ago
Source: Own?
Dear Mathlinkers,

1. ABC a triangle
2. 0 the circumcircle
3. D the pole of BC wrt 0
4. B', C' the symmetrics of B, C wrt AC, AB
5. 1b, 1c the circumcircles of the triangles BB'D, CC'D
6. J the center of 1b
7. V the second point of intersection of DJ and 1c.

Prove : CV is perpendicular to BC.

Sincerely
Jean-Louis
0 replies
1 viewing
jayme
2 hours ago
0 replies
Probably a good lemma
Zavyk09   4
N 4 hours ago by Zavyk09
Source: found when solving exercises
Let $ABC$ be a triangle with circumcircle $\omega$. Arbitrary points $E, F$ on $AC, AB$ respectively. Circumcircle $\Omega$ of triangle $AEF$ intersects $\omega$ at $P \ne A$. $BE$ intersects $CF$ at $I$. $PI$ cuts $\Omega$ and $\omega$ at $K, L$ respectively. Construct parallelogram $KFRE$. Prove that $A, R, L$ are collinear.
4 replies
Zavyk09
Yesterday at 12:50 PM
Zavyk09
4 hours ago
shade from tub
QueenArwen   1
N 4 hours ago by mikestro
Source: 46th International Tournament of Towns, Senior O-Level P4, Spring 2025
There was a tub on the plane, with its upper base greater that the lower one. The tub was overturned. Prove that the area of its visible shade did decrease. (The tub is a frustum of a right circular cone: its bases are two discs in parallel planes, such that their centers lie on a line perpendicular to these planes. The visible shade is the total shade besides the shade under the tub. Consider the sun rays as parallel.)
1 reply
QueenArwen
Mar 11, 2025
mikestro
4 hours ago
Inequality
Sappat   10
N 4 hours ago by iamnotgentle
Let $a,b,c$ be real numbers such that $a^2+b^2+c^2=1$. Prove that
$\frac{a^2}{1+2bc}+\frac{b^2}{1+2ca}+\frac{c^2}{1+2ab}\geq\frac{3}{5}$
10 replies
Sappat
Feb 7, 2018
iamnotgentle
4 hours ago
ISI UGB 2025 P4
SomeonecoolLovesMaths   9
N 4 hours ago by nyacide
Source: ISI UGB 2025 P4
Let $S^1 = \{ z \in \mathbb{C} \mid |z| =1 \}$ be the unit circle in the complex plane. Let $f \colon S^1 \longrightarrow S^2$ be the map given by $f(z) = z^2$. We define $f^{(1)} \colon = f$ and $f^{(k+1)} \colon = f \circ f^{(k)}$ for $k \geq 1$. The smallest positive integer $n$ such that $f^{(n)}(z) = z$ is called the period of $z$. Determine the total number of points in $S^1$ of period $2025$.
(Hint : $2025 = 3^4 \times 5^2$)
9 replies
SomeonecoolLovesMaths
May 11, 2025
nyacide
4 hours ago
Self-evident inequality trick
Lukaluce   9
N 5 hours ago by sqing
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let $x, y$, and $z$ be positive real numbers, such that $x^2 + y^2 + z^2 = 3$. Prove the inequality
\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]When does the equality hold?
9 replies
Lukaluce
Yesterday at 3:34 PM
sqing
5 hours ago
Prove n is square-free given divisibility condition
CatalanThinker   1
N 5 hours ago by CatalanThinker
Source: 1995 Indian Mathematical Olympiad
Let \( n \) be a positive integer such that \( n \) divides the sum
\[
1 + \sum_{i=1}^{n-1} i^{n-1}.
\]Prove that \( n \) is square-free.
1 reply
CatalanThinker
6 hours ago
CatalanThinker
5 hours ago
Integral inequality with differentiable function
Ciobi_   3
N Apr 9, 2025 by Levieee
Source: Romania NMO 2025 12.2
Let $f \colon [0,1] \to \mathbb{R} $ be a differentiable function such that its derivative is an integrable function on $[0,1]$, and $f(1)=0$. Prove that \[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]
3 replies
Ciobi_
Apr 2, 2025
Levieee
Apr 9, 2025
Integral inequality with differentiable function
G H J
G H BBookmark kLocked kLocked NReply
Source: Romania NMO 2025 12.2
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Ciobi_
28 posts
#1
Y by
Let $f \colon [0,1] \to \mathbb{R} $ be a differentiable function such that its derivative is an integrable function on $[0,1]$, and $f(1)=0$. Prove that \[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]
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MS_asdfgzxcvb
71 posts
#2 • 1 Y
Y by ehuseyinyigit
Using IBP, \(\displaystyle \int\textstyle xf=\displaystyle \int\textstyle \frac{-x^2f'}2\),
[asy]usepackage("amsmath, amssymb, tikz, tikz-cd");
label("\begin{tikzcd}[ampersand replacement=\&]
\displaystyle\int\scriptstyle x^2\displaystyle\int \scriptstyle(xf')^2\ \ge\ \left(\displaystyle\int\scriptstyle x^2f'\right)^2\&\&\displaystyle\int \scriptstyle(xf')^2\ \ge\ 3\left(\displaystyle\int\scriptstyle x^2f'\right)^2  
\arrow[Rightarrow, from=1-1, to=1-3]
\end{tikzcd}");[/asy]
so
[asy]usepackage("amsmath, amssymb, tikz, tikz-cd");
label("\begin{tikzcd}[ampersand replacement=\&]
\hspace{1pt}\&\&\color{blue}\displaystyle\int \scriptstyle(xf')^2\ \ge\ 12\left(\displaystyle\int\scriptstyle xf\right)^2.      
\arrow[Rightarrow, blue, from=1-1, to=1-3]
\end{tikzcd}");[/asy]
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Fibonacci_math
50 posts
#3
Y by
Easy one...

Note that using integration by parts, we get $$\int_0^1 xf(x) \ dx = \left[f(x)\frac{x^2}{2}\right]_0^1 - \int_0^1 f'(x)\frac{x^2}{2} \ dx=-\int_0^1 f'(x)\frac{x^2}{2} \ dx$$So, we need to show
$$\int_0^1 (xf'(x))^2 \ dx\ge 12 \left(\int_0^1 xf(x) \ dx\right)^2=12\left(\int_0^1 f'(x)\frac{x^2}{2} \ dx\right)^2=3\left(\int_0^1 x^2f'(x) \ dx\right)^2$$$$\iff \frac{1}{3}\left(\int_0^1 (xf'(x))^2 \ dx\right)\ge \left(\int_0^1 x^2f'(x) \ dx\right)^2$$$$\iff \left(\int_0^1 x^2 \ dx\right)\left(\int_0^1 (xf'(x))^2 \ dx\right)\ge \left(\int_0^1 x^2f'(x) \ dx\right)^2$$which is just Cauchy Schwarz inequality.
https://external-preview.redd.it/Vt8un6DvXelUjrQPqHsJXaIijhQIMDRU50RjKVXe2JM.jpg?auto=webp&s=fbd4da7e4d2893906b86cebbff64976f0e1c03e2

@below, thanks for pointing out the typo.
This post has been edited 1 time. Last edited by Fibonacci_math, Apr 9, 2025, 10:08 PM
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Levieee
243 posts
#4
Y by
same solution as the previous two
applying IBP on $\int_{0}^{1}xf$
we get
$ \int_{0}^{1} xf=\displaystyle \int_{0}^{1}\textstyle \frac{-x^2f'}2$

now it's equivalent to proof that
$\int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left(\int_{0}^{1}\textstyle \frac{-x^2f'}2\right)^2$
$\iff \frac{1}{3} \int_0^1 (x f'(x))^2 \, dx \geq \left( \int_0^1 x^2 f'(x) \, dx \right)^2$
$\iff \int_{0}^{1}x^{2} \int_0^1 (x f'(x))^2 \, dx \geq \left( \int_0^1 x^2 f'(x) \, dx \right)^2$
which is CS
https://static.wikia.nocookie.net/minecraft_gamepedia/images/e/eb/Plains_Baby_Villager_Base_JE2.png/revision/latest/scale-to-width/360?cb=20220612221105
$\mathbb{QED}$ $\blacksquare$
@above i think u made a typo by not putting $^{2}$ on the $\text{LHS}$ its a typo yes but that messes up the CS inequality
This post has been edited 4 times. Last edited by Levieee, Apr 9, 2025, 9:44 PM
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