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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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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
intersection of lines through midpoints are met at perpendicular bisector
azzam2912   2
N a few seconds ago by Giant_PT
Given a triangle $\triangle ABC$. The incircle of triangle $\triangle ABC$ is tangent to the sides $BC, CA,$ and $AB$ at points $D, E,$ and $F$, respectively. Let $X$ and $Y$ be the midpoints of $BD$ and $DC$, respectively. Let $P$ and $Q$ be the midpoints of $BF$ and $CE$, respectively. Prove that $XP$ and $YQ$ intersect on the perpendicular bisector of side $BC$.

notes. : does anyone knows from which source this question comes from?
2 replies
azzam2912
an hour ago
Giant_PT
a few seconds ago
All even perfect numbers >6 equal x^3 + y^3 + z^3
TUAN2k8   2
N a few seconds ago by Lufin
Source: 2025 VIASM summer challenge P5
Let $n$ be an even positive integer greater than 6 such that $2n$ is equal to the sum of all distinct positive divisors of $n$. Prove that there exist distinct integers $x, y, z$ satisfying:
\[
x^3 + y^3 + z^3 = n.
\]
2 replies
1 viewing
TUAN2k8
Monday at 1:55 PM
Lufin
a few seconds ago
beautiful geometry
pnf   1
N 3 minutes ago by Seungjun_Lee
let $\triangle$ $ABC$ be a triangle with circumcenter $O$.$\ell$ is a fix line passing through $O$ and $p$ an arbitrary point on it . denote $\omega$ the circumcircle of pedal triangle of $p$. prove that $\omega$ passese through a fix point
1 reply
pnf
Yesterday at 4:31 PM
Seungjun_Lee
3 minutes ago
Interesting inequality
sqing   0
17 minutes ago
Source: Own
Let $ a,b,c \geq0,a^2 +b^2+c^2=1 . $ Prove that
$$  \frac{1}{2} \leq \frac{a+b+c}{ab+bc+ca+\frac{217}{90}abc+2} \leq\frac{2\sqrt 2}{5} $$$$  \frac{18(18\sqrt 3-5)}{947} \leq \frac{a+b+c}{ab+bc+ca+\frac{5}{2}abc+2} \leq\frac{2\sqrt 2}{5} $$
0 replies
1 viewing
sqing
17 minutes ago
0 replies
f(x^2+x+1)=f(x)f(x+1)
MathMaxGreat   0
21 minutes ago
Source: 100 algebra problems
Problem1:Find all $f(x)\in \mathbb{C} [x]$, s.t. $\forall x\in \mathbb{C},f(x^2+x+1)=f(x)f(x+1)$
0 replies
MathMaxGreat
21 minutes ago
0 replies
minimize
Butterfly   0
an hour ago

Minimize $a^2+b^2$ on $\frac{a}{b}+\frac{b}{a}+2ab=a^2-b^2$ and $a,b>0$.
0 replies
Butterfly
an hour ago
0 replies
Interesting inequality
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b>0. $ Prove that
$$\frac{(a^2-a+b+k)(b^2-b+a+k)}{a+b}    \geq \frac{8k\sqrt {3k}}{9}$$Where $ k>0. $
1 reply
1 viewing
sqing
Yesterday at 2:56 AM
sqing
an hour ago
Miquel circles and a beautiful similarity
pohoatza   52
N an hour ago by Kempu33334
Source: IMO Shortlist 2006, Geometry 9, AIMO 2007, TST 2, P3
Points $ A_{1}$, $ B_{1}$, $ C_{1}$ are chosen on the sides $ BC$, $ CA$, $ AB$ of a triangle $ ABC$ respectively. The circumcircles of triangles $ AB_{1}C_{1}$, $ BC_{1}A_{1}$, $ CA_{1}B_{1}$ intersect the circumcircle of triangle $ ABC$ again at points $ A_{2}$, $ B_{2}$, $ C_{2}$ respectively ($ A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$). Points $ A_{3}$, $ B_{3}$, $ C_{3}$ are symmetric to $ A_{1}$, $ B_{1}$, $ C_{1}$ with respect to the midpoints of the sides $ BC$, $ CA$, $ AB$ respectively. Prove that the triangles $ A_{2}B_{2}C_{2}$ and $ A_{3}B_{3}C_{3}$ are similar.
52 replies
pohoatza
Jun 28, 2007
Kempu33334
an hour ago
100 Algebra Problems
MathMaxGreat   0
an hour ago
Source: Own
I had a plan about posting 100 algebra problem, difficulty ranging from $ChinaSecond Round 1$ to $IMO2$, for us to train, I’ll post about 1~2 problems a day, after that, A collection of these problems will be made.
I hope the problems are not old ones.
0 replies
MathMaxGreat
an hour ago
0 replies
Quadrilateral ABCD such that AC+AD=BC+BD
WakeUp   16
N an hour ago by shendrew7
Source: Canadian MO 2012 #3
Let $ABCD$ be a convex quadrilateral and let $P$ be the point of intersection of $AC$ and $BD$. Suppose that $AC+AD=BC+BD$. Prove that the internal angle bisectors of $\angle ACB$, $\angle ADB$ and $\angle APB$ meet at a common point.
16 replies
WakeUp
May 4, 2012
shendrew7
an hour ago
Add d or Divide by a
MarkBcc168   29
N 2 hours ago by eg4334
Source: ISL 2022 N3
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
29 replies
MarkBcc168
Jul 9, 2023
eg4334
2 hours ago
Periodic sequence with positive integers
thdnder   43
N 2 hours ago by sami1618
Source: IMO 2024 P3
Let $a_1, a_2, a_3, \dots$ be an infinite sequence of positive integers, and let $N$ be a positive integer. Suppose that, for each $n > N$, $a_n$ is equal to the number of times $a_{n-1}$ appears in the list $a_1, a_2, \dots, a_{n-1}$.

Prove that at least one of the sequence $a_1, a_3, a_5, \dots$ and $a_2, a_4, a_6, \dots$ is eventually periodic.

(An infinite sequence $b_1, b_2, b_3, \dots$ is eventually periodic if there exist positive integers $p$ and $M$ such that $b_{m+p} = b_m$ for all $m \ge M$.)
43 replies
thdnder
Jul 16, 2024
sami1618
2 hours ago
Avoid prime sum
NicoN9   4
N 2 hours ago by AbbyWong
Source: Based on https://atcoder.jp/contests/arc149/tasks/arc149_c?lang=en
An integer $n\ge 3$ and a $n\times n$ grid are given. We write down one integer to each of the square of the grid so that each of integer $1$ to $n^2$ are written exactly once in the grid, and no two adjacent square have sum prime.

Prove that this is always possible.
4 replies
NicoN9
Jul 25, 2025
AbbyWong
2 hours ago
Density in Factorial Congruence
steven_zhang123   1
N 2 hours ago by zqy648
Source: 2025 Jul-谜之竞赛 Round 2 P3
Let \(p\) be a prime. For a positive integer \(m\), denote \(\nu_p(m)\) as the unique nonnegative integer \(k\) such that \(p^k \mid m\) but \(p^{k+1} \nmid m\).
Prove that for any real \(\varepsilon > 0\), there exists a positive integer \(N\) such that for all positive integers \(n \geq N\), at least \(\left( \frac{1}{p-1} - \varepsilon \right) \cdot n\) positive integers \(m\) in \(1, 2, \cdots, n\) satisfy
\[
\frac{m!}{p^{\nu_p(m!)}} \equiv 1 \pmod{p}.
\]Proposed by Dong Zhenyu
1 reply
+1 w
steven_zhang123
Jul 27, 2025
zqy648
2 hours ago
Computing functions
BBNoDollar   3
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 $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )
3 replies
BBNoDollar
May 21, 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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alexheinis
10774 posts
#2
Y by
Write $A:=\begin{pmatrix} a&b\\c&d\end{pmatrix},B:=\begin{pmatrix} 1&0\\1&1\end{pmatrix}$ and suppose that $A^n=kB^n$ where $k$ is some constant and where $n>1$ is a fixed positive integer. We want to show that $A=tB$ for some $t$. Since $A^n$ has a positive entry at position $(1,1)$, we have $k>0$ and normalising by a positive factor we assume $A^n=B^n$.
The char equation for $A$ reads $(t-a)(t-d)=bc\iff t^2-(a+d)t+(ad-bc)=0$ with $D=(a+d)^2-4(ad-bc)=(a-d)^2+4bc\ge 0$. Hence $A$ has real eigenvalues $\lambda,\mu$ and $\{\lambda^n,\mu^n\}=\{1,1\}$ as a multiset, since $B^n$ has a multiple eigenvalue 1. Hence $|\lambda|=|\mu|=1$ and since $\lambda+\mu={\rm Tr}(A)=a+d>0$ we have $\lambda=\mu=1$. Hence $A^2=2A-I, B^2=2B-I$.
Then $kA+lI=A^n=B^n=kB+lI$ where $k,l$ are constants depending on $n$. Hence $k(A-B)=0$ and note that $k\not=0$. Either by direct computation or by noting that $B^n$ is not a multiple of $I$.
Hence $A=B$.

@below: it is standard procedure to represent $f(x)={{ax+b}\over {cx+d}}$ by the matrix $A:=\begin{pmatrix} a&b\\c&d\end{pmatrix}$. A real number $x$ is represented by the vector ${x\choose 1}$, the point at $\infty$ is represented by ${1\choose 0}$ and we have $f(x)=Ax$ on the projective line. Indeed $f(\infty)=a/c$. Then $f^n$ is represented by $A^n$ and we do this because it is easier to multiply matrices than to calculate the composition $f_n:=f\circ \cdots \circ f$ directly. But we have to note that matrices that are multiples of eachother describe the same linear fractional transformation.
But to answer your question: the motivation is that I have seen method before and it works.
This post has been edited 1 time. Last edited by alexheinis, May 22, 2025, 6:32 PM
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youochange
200 posts
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alexheinis wrote:
Write $A:=\begin{pmatrix} a&b\\c&d\end{pmatrix},B:=\begin{pmatrix} 1&0\\1&1\end{pmatrix}$ and suppose that $A^n=kB^n$ where $k$ is some constant and where $n>1$ is a fixed positive integer. We want to show that $A=tB$ for some $t$. Since $A^n$ has a positive entry at position $(1,1)$, we have $k>0$ and normalising by a positive factor we assume $A^n=B^n$.
The char equation for $A$ reads $(t-a)(t-d)=bc\iff t^2-(a+d)t+(ad-bc)=0$ with $D=(a+d)^2-4(ad-bc)=(a-d)^2+4bc\ge 0$. Hence $A$ has real eigenvalues $\lambda,\mu$ and $\{\lambda^n,\mu^n\}=\{1,1\}$ as a multiset, since $B^n$ has a multiple eigenvalue 1. Hence $|\lambda|=|\mu|=1$ and since $\lambda+\mu={\rm Tr}(A)=a+d>0$ we have $\lambda=\mu=1$. Hence $A^2=2A-I, B^2=2B-I$.
Then $kA+lI=A^n=B^n=kB+lI$ where $k,l$ are constants depending on $n$. Hence $k(A-B)=0$ and note that $k\not=0$. Either by direct computation or by noting that $B^n$ is not a multiple of $I$.
Hence $A=B$.

Sir, can you pls explain what was the motivation behind your solution? :D
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wh0nix
27 posts
#4 • 1 Y
Y by youochange
Hint
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