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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

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0 replies
jwelsh
Jul 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
Circumcenter on angle bisector
DottedCaculator   8
N a minute ago by Korean_fish_Kaohsiung
Source: 2025 ELMO Shortlist G6
Let $ABC$ be a triangle with incenter $I$, and let the midpoint of $BC$ be $M$. Ray $MI$ intersects $AB$ and $AC$ at $P$ and $Q$. If $N$ is the midpoint of major arc $BC$, prove that the circumcenter of triangle $NPQ$ lies on $AI$.

Rohan Bodke
8 replies
DottedCaculator
Jun 30, 2025
Korean_fish_Kaohsiung
a minute ago
AOPS MO discussions
MathMaxGreat   0
11 minutes ago
WOW let’s meet here:https://artofproblemsolving.com/community/c4388216
0 replies
MathMaxGreat
11 minutes ago
0 replies
AOPS MO Introduce
MathMaxGreat   16
N 14 minutes ago by TigerOnion
$AOPS MO$

Problems: post it as a private message to me or @jerryZYang, please post it in $LATEX$ and have answers

6 Problems for two rounds, easier than $IMO$

If you want to do the problems or be interested, reply ’+1’
Want to post a problem reply’+2’ and message me
Want to be in the problem selection committee, reply’+3’
16 replies
MathMaxGreat
5 hours ago
TigerOnion
14 minutes ago
concyclic
Jumbler   25
N an hour ago by Learning11
Source: Chinese Western Mathematical Olympiad 2006, Problem 6
$AB$ is a diameter of the circle $O$, the point $C$ lies on the line $AB$ produced. A line passing though $C$ intersects with the circle $O$ at the point $D$ and $E$. $OF$ is a diameter of circumcircle $O_{1}$ of $\triangle BOD$. Join $CF$ and produce, cutting the circle $O_{1}$ at $G$. Prove that points $O,A,E,G$ are concyclic.
25 replies
Jumbler
Nov 7, 2006
Learning11
an hour ago
inequalities
pennypc123456789   4
N an hour ago by SunnyEvan
If $a,b,c$ are positive real numbers, then
$$
\frac{a + b}{a + 7b + c} + \dfrac{b + c}{b + 7c + a}+\dfrac{c + a}{c + 7a + b} \geq \dfrac{2}{3}$$
we can generalize this problem
4 replies
pennypc123456789
Apr 17, 2025
SunnyEvan
an hour ago
Angelic tetrahedrons
v_Enhance   6
N an hour ago by numbersandnumbers
Source: USA TSTST 2025/5
A tetrahedron $ABCD$ is said to be angelic if it has nonzero volume and satisfies \[ \begin{aligned} \angle BAC + \angle CAD + \angle DAB &= \angle ABC + \angle CBD + \angle DBA, \\ \angle ACB + \angle BCD + \angle DCA &= \angle ADB + \angle BDC + \angle CDA. \end{aligned} \]Across all angelic tetrahedrons, what is the maximum number of distinct lengths that could appear in the set $\{AB,AC,AD,BC,BD,CD\}$?

Karthik Vedula
6 replies
v_Enhance
Jul 1, 2025
numbersandnumbers
an hour ago
Number theory - Iran
soroush.MG   35
N an hour ago by NicoN9
Source: Iran MO 2017 - 2nd Round - P1
a) Prove that there doesn't exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: gcd(a_i+j,a_j+i)=1$

b) Let $p$ be an odd prime number. Prove that there exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: p \not | gcd(a_i+j,a_j+i)$
35 replies
soroush.MG
Apr 20, 2017
NicoN9
an hour ago
How about an AOPS MO?
MathMaxGreat   36
N 2 hours ago by JerryZYang
I am planning to make a $APOS$ $MO$, we can post new and original problems, my idea is to make an competition like $IMO$, 6 problems for 2 rounds
Any idea and plans?
36 replies
MathMaxGreat
Yesterday at 2:37 AM
JerryZYang
2 hours ago
2^n + 3^n is never a perfect cube
parmenides51   4
N 2 hours ago by TigerOnion
Source: JBMO 2008 Shortlist N10
Prove that $2^n + 3^n$ is not a perfect cube for any positive integer $n$.
4 replies
parmenides51
Oct 14, 2017
TigerOnion
2 hours ago
Choose lattice points so that no four points are cyclic
rkm0959   11
N 2 hours ago by Treblax08
Source: 2016 Final Korean Mathematical Olympiad P2
Two integers $n, k$ satisfies $n \ge 2$ and $k \ge \frac{5}{2}n-1$.
Prove that whichever $k$ lattice points with $x$ and $y$ coordinate no less than $1$ and no more than $n$ we pick, there must be a circle passing through at least four of these points.
11 replies
rkm0959
Mar 19, 2016
Treblax08
2 hours ago
Interesting Spiral
VitaPretor   2
N 3 hours ago by Bread10
a) We start at $(0,0)$ and walk $400$ feet north and turn $90$ degrees to the right.
We then walk 75% of $400$ or $300$ feet east and turn $90$ degrees to the right.
We next walk 75% of $300$ or $225$ feet south and turn $90$ degrees to the right.
We repeat the process indefinitely of walking 75% of the distance that we last walked and turning $90$ degrees to the right forming a spiral. What are the exact coordinates we approach after repeating the process indefinitely?

b) We start at $(0,0)$ and walk $x$ feet north and turn $90$ degrees to the right.
We then walk $x * y$ feet to the east and turn $90$ degrees to the right.
We next walk $x * y^2$ feet south and turn $90$ degrees to the right.
We repeat the process indefinitely of walking $y$ times the distance that we last walked and turning $90$ degrees to the right forming a spiral.
Assume $x$ > $0$ and $0$ < $y$ < $1$. If the point we eventually approach is $(50,60)$ find the ordered pair $(x,y)$.
2 replies
VitaPretor
Yesterday at 12:45 AM
Bread10
3 hours ago
Easy with 3 var and parameter
mihaig   1
N 4 hours ago by pooh123
Source: Own
Find the smallest real constant $K$ such that
$$18+3abc\geq7\left(ab+bc+ca\right)$$for all $a,b,c\geq K$ satisfying $a+b+c=3.$
1 reply
mihaig
Yesterday at 6:23 AM
pooh123
4 hours ago
Infimum of decreasing sequence b_n/n^2
a1267ab   36
N 4 hours ago by brainfertilzer
Source: USA Winter TST for IMO 2020, Problem 1 and TST for EGMO 2020, Problem 3, by Carl Schildkraut and Milan Haiman
Choose positive integers $b_1, b_2, \dotsc$ satisfying
\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of the sequence $(b_n)$?

Carl Schildkraut and Milan Haiman
36 replies
a1267ab
Dec 16, 2019
brainfertilzer
4 hours ago
IMO Genre Predictions
ohiorizzler1434   111
N 4 hours ago by ohiorizzler1434
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
111 replies
ohiorizzler1434
May 3, 2025
ohiorizzler1434
4 hours ago
Computing functions
BBNoDollar   8
N May 24, 2025 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 $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
8 replies
BBNoDollar
May 18, 2025
wh0nix
May 24, 2025
Computing functions
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G H BBookmark kLocked kLocked NReply
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BBNoDollar
15 posts
#1
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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 $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
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alinazarboland
172 posts
#2
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Here's a sketch of a method which solves every single mobius tranform problem I saw.
Let $z_1,z_2$ be the two complex roots of $f(z)=z$. Then, since a mobius transform is just a combination of shifting,scaling,rotating, and inversion, for any complex number $z$ we have:
$$(z , \infty ; z_1,z_2) = (f(z) , \frac{a}{c} ; z_1,z_2)$$If you write this $n$ times you'd get:
$$k^n .\frac{z-z_1}{z-z_2} = \frac{f_n - z_1}{f_n - z_2}$$Where $k = \frac{a/c - z_1}{a/c - z_2}$.Now let $f_n(x) = \frac{x}{1 + nx}$ for some $n$. One can easily get $k^n=1$(by comparing the coefficient of $x$ in the respective polynomial identity) and so $x_1=x_2$(comparing $x^2$s).
Now, $x_1=x_2$ means we have a double root for $f(x)=x$ and delta=0 so $(d-a)^2+4bc=0$. Combining with the fact that $x_1,x_2$ are fix points of every $f_k$ , we'll get $(n-1)^2+0=0$ and $n=1$
This post has been edited 2 times. Last edited by alinazarboland, May 18, 2025, 7:42 PM
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alinazarboland
172 posts
#3
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Here are two old problems one from $2012$ IMC and one from Iranian Olympiad which are trivial with this method
https://artofproblemsolving.com/community/c7h491145p2754513
https://artofproblemsolving.com/community/c6h368215p2026678
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BBNoDollar
15 posts
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alinazarboland wrote:
Here's a sketch of a method which solves every single mobius tranform problem I saw.
Let $z_1,z_2$ be the two complex roots of $f(z)=z$. Then, since a mobius transform is just a combination of shifting,scaling,rotating, and inversion, for any complex number $z$ we have:
$$(z , \infty ; z_1,z_2) = (f(z) , \frac{a}{c} ; z_1,z_2)$$If you write this $n$ times you'd get:
$$k^n .\frac{z-z_1}{z-z_2} = \frac{f_n - z_1}{f_n - z_2}$$Where $k = \frac{a/c - z_1}{a/c - z_2}$.Now let $f_n(x) = \frac{x}{1 + nx}$ for some $n$. One can easily get $k^n=1$(by comparing the coefficient of $x$ in the respective polynomial identity) and so $x_1=x_2$(comparing $x^2$s).
Now, $x_1=x_2$ means we have a double root for $f(x)=x$ and delta=0 so $(d-a)^2+4bc=0$. Combining with the fact that $x_1,x_2$ are fix points of every $f_k$ , we'll get $(n-1)^2+0=0$ and $n=1$

Thank you very much, i appreciate this solution ! I can understand it, but i need a 9th grade solution. I solved the ''reciprocal'' implication by induction, now i need to demonstrate the ''direct'' one. Can you or anyone help me ?
This post has been edited 1 time. Last edited by BBNoDollar, May 18, 2025, 10:15 PM
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ICE_CNME_4
22 posts
#5
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Bumping this
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ICE_CNME_4
22 posts
#6
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Bump. Bump
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BBNoDollar
15 posts
#7
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BUMPING for 9th grade solution
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ICE_CNME_4
22 posts
#8
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Someone for this?
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wh0nix
27 posts
#9
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Hint
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