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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.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
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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
ISI UGB 2025 P4
SomeonecoolLovesMaths   3
N 6 minutes ago by SomeonecoolLovesMaths
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 : $2025 = 3^4 \times 5^2$)
3 replies
+2 w
SomeonecoolLovesMaths
4 hours ago
SomeonecoolLovesMaths
6 minutes ago
ISI UGB 2025 P7
SomeonecoolLovesMaths   7
N 9 minutes ago by SomeonecoolLovesMaths
Source: ISI UGB 2025 P7
Consider a ball that moves inside an acute-angled triangle along a straight line, unit it hits the boundary, which is when it changes direction according to the mirror law, just like a ray of light (angle of incidence = angle of reflection). Prove that there exists a triangular periodic path for the ball, as pictured below.

IMAGE
7 replies
SomeonecoolLovesMaths
3 hours ago
SomeonecoolLovesMaths
9 minutes ago
Interesting inequalities
sqing   4
N 14 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 , (a+k )(b+c)=k+1.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq  \frac{2k-3+2\sqrt{k+1}}{3k-1}$$Where $ k\geq \frac{2}{3}.$
Let $ a,b,c\geq 0 , (a+1)(b+c)=2.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq 2\sqrt{2}-1$$Let $ a,b,c\geq 0 , (a+3)(b+c)=4.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq  \frac{7}{4}$$Let $ a,b,c\geq 0 , (3a+2)(b+c)= 5.$ Prove that
$$\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}\geq  \frac{2(2\sqrt{15}-5)}{3}$$
4 replies
1 viewing
sqing
Yesterday at 1:29 PM
sqing
14 minutes ago
R to R, with x+f(xy)=f(1+f(y))x
NicoN9   4
N 23 minutes ago by CM1910
Source: Own.
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that\[
x+f(xy)=f(1+f(y))x
\]for all $x, y\in \mathbb{R}$.
4 replies
NicoN9
Today at 8:52 AM
CM1910
23 minutes ago
JBMO Shortlist 2021 G5
Lukaluce   7
N 41 minutes ago by africanboy
Source: JBMO Shortlist 2021
Let $ABC$ be an acute scalene triangle with circumcircle $\omega$. Let $P$ and $Q$ be interior points of the sides $AB$ and $AC$, respectively, such that $PQ$ is parallel to $BC$. Let $L$ be a point on $\omega$ such that $AL$ is parallel to $BC$. The segments $BQ$ and $CP$ intersect at $S$. The line $LS$ intersects $\omega$ at $K$. Prove that $\angle BKP = \angle CKQ$.

Proposed by Ervin Macić, Bosnia and Herzegovina
7 replies
Lukaluce
Jul 2, 2022
africanboy
41 minutes ago
n+c divides a^n+b^n+n for certain n
v_Enhance   29
N 42 minutes ago by alexanderhamilton124
Source: ELMO 2014 Shortlist N7, by Evan Chen
Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$.

Proposed by Evan Chen
29 replies
v_Enhance
Jul 24, 2014
alexanderhamilton124
42 minutes ago
thank you !
Piwbo   0
an hour ago
In a conference with $n$ attendees $(n\ge 2 , n \in \mathbb{N})$ a group of $ k$ people is called a "beautiful$ k$-group" if any two people within that $k$ people know each other. Assume that every beautiful $ k$-group has exactly one person in common, and there is no beautiful 5-group. Prove that there $\exists $ 2 people such that if they leave the conference, there will be no beautiful 3-group remaining.
0 replies
Piwbo
an hour ago
0 replies
Asymmetric FE
sman96   15
N an hour ago by youochange
Source: BdMO 2025 Higher Secondary P8
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(xf(y)-y) + f(xy-x) + f(x+y) = 2xy$$for all $x, y \in \mathbb{R}$.
15 replies
sman96
Feb 8, 2025
youochange
an hour ago
Looks like power mean, but it is not
Nuran2010   4
N an hour ago by sqing
Source: Azerbaijan Al-Khwarizmi IJMO TST 2025
For $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2 \geq 3$,show that:

$\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$.
4 replies
Nuran2010
3 hours ago
sqing
an hour ago
ISI UGB 2025 P6
SomeonecoolLovesMaths   1
N an hour ago by ronitdeb
Source: ISI UGB 2025 P6
Let $\mathbb{N}$ denote the set of natural numbers, and let $\left( a_i, b_i \right)$, $1 \leq i \leq 9$, be nine distinct tuples in $\mathbb{N} \times \mathbb{N}$. Show that there are three distinct elements in the set $\{ 2^{a_i} 3^{b_i} \colon 1 \leq i \leq 9 \}$ whose product is a perfect cube.
1 reply
SomeonecoolLovesMaths
4 hours ago
ronitdeb
an hour ago
A bit tricky invariant with 98 numbers on the board.
Nuran2010   2
N 2 hours ago by Tung-CHL
Source: Azerbaijan Al-Khwarizmi IJMO TST 2025
The numbers $\frac{50}{1},\frac{50}{2},...\frac{50}{97},\frac{50}{98}$ are written on the board.In each step,two random numbers $a$ and $b$ are chosen and deleted.Then,the number $2ab-a-b-1$ is written instead.What will be the number remained on the board after the last step.
2 replies
Nuran2010
3 hours ago
Tung-CHL
2 hours ago
ISI UGB 2025 P2
SomeonecoolLovesMaths   1
N 2 hours ago by Project_Donkey_into_M4
Source: ISI UGB 2025 P2
If the interior angles of a triangle $ABC$ satisfy the equality, $$\sin ^2 A + \sin ^2 B + \sin^2  C = 2 \left( \cos ^2 A + \cos ^2 B + \cos ^2 C \right),$$prove that the triangle must have a right angle.
1 reply
SomeonecoolLovesMaths
4 hours ago
Project_Donkey_into_M4
2 hours ago
hard inequality
moldovan   2
N 2 hours ago by ririgggg
Source: Austria 1987
Let $ x_1,...,x_n$ be positive real numbers. Prove that:

$ \displaystyle\sum_{k=1}^{n}x_k+\sqrt{\displaystyle\sum_{k=1}^{n}x_k^2} \le \frac{n+\sqrt{n}}{n^2} \left( \displaystyle\sum_{k=1}^{n} \frac{1}{x_k} \right) \left( \displaystyle\sum_{k=1}^{n} x_k^2 \right).$
2 replies
moldovan
Jul 8, 2009
ririgggg
2 hours ago
ISI UGB 2025 P5
SomeonecoolLovesMaths   3
N 2 hours ago by lakshya2009
Source: ISI UGB 2025 P5
Let $a,b,c$ be nonzero real numbers such that $a+b+c \neq 0$. Assume that $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a+b+c}$$Show that for any odd integer $k$, $$\frac{1}{a^k} + \frac{1}{b^k} + \frac{1}{c^k} = \frac{1}{a^k+b^k+c^k}.$$
3 replies
SomeonecoolLovesMaths
4 hours ago
lakshya2009
2 hours ago
true or false ?
SunnyEvan   5
N Apr 25, 2025 by SunnyEvan
Let $ a,b,c \geq 0 $ Prove that :
$$ \frac{3k}{k^4+k+1} \leq \frac{ka}{a+k^4b+kc}+\frac{kb}{b+k^4c+ka}+\frac{kc}{c+k^4a+kb} \leq k $$Where $ k \geq 1 $
5 replies
SunnyEvan
Apr 20, 2025
SunnyEvan
Apr 25, 2025
true or false ?
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SunnyEvan
118 posts
#1
Y by
Let $ a,b,c \geq 0 $ Prove that :
$$ \frac{3k}{k^4+k+1} \leq \frac{ka}{a+k^4b+kc}+\frac{kb}{b+k^4c+ka}+\frac{kc}{c+k^4a+kb} \leq k $$Where $ k \geq 1 $
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SunnyEvan
118 posts
#4
Y by
Bump for it. :)
This post has been edited 1 time. Last edited by SunnyEvan, Apr 20, 2025, 12:27 PM
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GeoMorocco
44 posts
#5
Y by
SunnyEvan wrote:
Let $ a,b,c \geq 0 $ Prove that :
$$ \frac{3k}{k^4+k+1} \leq \frac{ka}{a+k^4b+kc}+\frac{kb}{b+k^4c+ka}+\frac{kc}{c+k^4a+kb} \leq k $$Where $ k \geq 1 $

The Right side is easy:
$$ \frac{ka}{a+k^4b+kc}+\frac{kb}{b+k^4c+ka}+\frac{kc}{c+k^4a+kb} \leq \frac{ka}{a+b+c}+\frac{kb}{b+c+a}+\frac{kc}{c+a+b} =k $$
For the Left side, you can use Cauchy-Schwarz, and let $t = \frac{a^2+b^2+c^2}{ab+bc+ca} \geq 1$. and we only need to prove:
$$\frac{(a+b+c)^2}{a^2+b^2+c^2+(k^4+k)(ab+bc+ca)} = \frac{t+2}{t+k^4+k}\geq \frac{3}{k^4+k+1}$$Or:
$$\frac{(k-1)(t-1)(k^3+k^2+k+1)}{(k^4+k+1)(k^4+k+t)} \geq 0$$which is true.
This post has been edited 4 times. Last edited by GeoMorocco, Apr 20, 2025, 12:37 PM
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SunnyEvan
118 posts
#6
Y by
Let $ a,b,c \geq 0 $ Prove that :
$$ \frac{a}{ka+k^2b+c}+\frac{b}{kb+k^2c+a}+\frac{c}{kc+k^2a+b} \leq \frac{1}{k} \leq   \frac{a}{ka+(2k-1)b+c}+\frac{b}{kb+(2k-1)+a}+\frac{c}{kc+(2k-1)a+b} $$Where $ k \geq 1 $
This post has been edited 3 times. Last edited by SunnyEvan, Apr 25, 2025, 1:21 PM
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SunnyEvan
118 posts
#7
Y by
GeoMorocco wrote:
SunnyEvan wrote:
Let $ a,b,c \geq 0 $ Prove that :
$$ \frac{3k}{k^4+k+1} \leq \frac{ka}{a+k^4b+kc}+\frac{kb}{b+k^4c+ka}+\frac{kc}{c+k^4a+kb} \leq k $$Where $ k \geq 1 $

The Right side is easy:
$$ \frac{ka}{a+k^4b+kc}+\frac{kb}{b+k^4c+ka}+\frac{kc}{c+k^4a+kb} \leq \frac{ka}{a+b+c}+\frac{kb}{b+c+a}+\frac{kc}{c+a+b} =k $$
For the Left side, you can use Cauchy-Schwarz, and let $\frac{a^2+b^2+c^2}{ab+bc+ca} \geq 1$. and we only need to prove:
$$\frac{(a+b+c)^2}{a^2+b^2+c^2+(k^4+k)(ab+bc+ca)} = \frac{t+2}{t+k^4+k}\geq \frac{3}{k^4+k+1}$$which is true:
$$\frac{(k-1)(t-1)(k^3+k^2+k+1)}{(k^4+k+1)(k^4+k+t)} \geq 0$$which is true.

Thanks for your help GeoMorocco.
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SunnyEvan
118 posts
#8
Y by
SunnyEvan wrote:
Let $ a,b,c \geq 0 $ Prove that :
$$ \frac{a}{ka+k^2b+c}+\frac{b}{kb+k^2c+a}+\frac{c}{kc+k^2a+b} \leq \frac{1}{k} \leq   \frac{a}{ka+(2k-1)b+c}+\frac{b}{kb+(2k-1)+a}+\frac{c}{kc+(2k-1)a+b} $$Where $ k \geq 1 $

How about this?
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