Plan ahead for the next school year. Schedule your class today!

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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
and train with the best! Please note that early bird pricing ends August 19th!
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.

Our full course list for upcoming classes is below:
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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
Have fun
centslordm   7
N 8 minutes ago by Cats_on_a_computer
Source: Instagram
Compute \[\prod^\infty_{k=2} \sqrt[2^k]{1 - \frac 1k - \frac 1{k^2} + \frac1{k^3}}.\]
7 replies
centslordm
Today at 5:40 AM
Cats_on_a_computer
8 minutes ago
INMO 2019 P1
div5252   61
N an hour ago by SomeonecoolLovesMaths
Let $ABC$ be a triangle with $\angle{BAC} > 90$. Let $D$ be a point on the segment $BC$ and $E$ be a point on line $AD$ such that $AB$ is tangent to the circumcircle of triangle $ACD$ at $A$ and $BE$ is perpendicular to $AD$. Given that $CA=CD$ and $AE=CE$. Determine $\angle{BCA}$ in degrees.
61 replies
div5252
Jan 20, 2019
SomeonecoolLovesMaths
an hour ago
FE Tree!
Primeniyazidayi   35
N an hour ago by MathsII-enjoy
Hello guys!
I want to start a FE tree.One will post a FE equation,and others will try to solve.There will be two problems to solve.If you have solved one of them,post a new one.If two,then post two FEs.
I start:
P1

P2
35 replies
Primeniyazidayi
Jun 12, 2025
MathsII-enjoy
an hour ago
Central sequences
EeEeRUT   17
N an hour ago by dgrozev
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.
17 replies
EeEeRUT
Apr 16, 2025
dgrozev
an hour ago
Inscribed Triangle and its Center of Mass
scarlet128   0
an hour ago
Given a circle $C$ and $3$ randomly selected points on $C$, call $M$ the center of mass of those $3$ points. What is the probability that $M$ is closer to $C$'s circumference than to its center?
0 replies
scarlet128
an hour ago
0 replies
IMSC 2024 - MOCK TEST 3
MathsII-enjoy   2
N an hour ago by MathsII-enjoy
Circles $\omega_1$, $\omega_2$ intersects at $P,K$. $XY$ is the common tangent of the two circles which is nearer to $P$, where $X$ is on $\omega_1$ and $Y$ is on $\omega_2$. $XP$ intersects $\omega_2$ for the second time in $C$ and $YP$ intersects $\omega_1$ again in $B$. Let $A$ be the intersection of $BX$ and $CY$. Let $Q$ be the second intersection point of the circumcircles of $ABC$ and $AXY$. Prove that $\widehat{QXA}=\widehat{QKP}$
2 replies
MathsII-enjoy
Jul 6, 2025
MathsII-enjoy
an hour ago
IMO 2014 Problem 2
v_Enhance   64
N 2 hours ago by fearsum_fyz
Source: 0
Let $n \ge 2$ be an integer. Consider an $n \times n$ chessboard consisting of $n^2$ unit squares. A configuration of $n$ rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k \times k$ square which does not contain a rook on any of its $k^2$ unit squares.
64 replies
v_Enhance
Jul 8, 2014
fearsum_fyz
2 hours ago
algerba 2025 Turkey National Olympiad First round
Razorrizelim   0
2 hours ago
Source: 2025 Turkey National Olympiad First round
Let \( m \) and \( n \) be integers. How many quadratic polynomials of the form
\( f(x) = x^2 + mx + n \) satisfy the condition \( f(f(360)) = 0 \)? \[ \textbf{(A)}\ 1 \quad \textbf{(B)}\ 18 \quad \textbf{(C)}\ 48 \quad \textbf{(D)}\ 60 \quad \textbf{(E)}\ \text{None} \]
0 replies
Razorrizelim
2 hours ago
0 replies
CWMI 2016 Problem 7
YanYau   7
N 2 hours ago by eulerleonhardfan
Source: CWMI
$ABCD$ is a cyclic quadrilateral, and $\angle BAC = \angle DAC$. $\astrosun I_1$ and $\astrosun I_2$ are the incircles of $\triangle ABD$ and $\triangle ADC$ respectively. Prove that one of the common external tangents of $\astrosun I_1$ and $\astrosun I_2$ is parallel to $BD$
7 replies
YanYau
Aug 19, 2016
eulerleonhardfan
2 hours ago
number theory 2025 Turkey National Olympiad First round
Razorrizelim   0
3 hours ago
Source: 2025 Turkey National Olympiad First round
Problem 2: How many positive integers \( n < 2025 \) are there such that the number \( 1^3 + 2^3 + \cdots + n^3 \) is divisible by 2025? \[ \textbf{(A)}\ 44 \quad \textbf{(B)}\ 89 \quad \textbf{(C)}\ 134 \quad \textbf{(D)}\ 179 \quad \textbf{(E)}\ 224 \]
0 replies
Razorrizelim
3 hours ago
0 replies
Functional Number Theory
Iveela   5
N 3 hours ago by cursed_tangent1434
Source: 2025 IRN-MNG Friendly Competition
Let $\mathbb{N}$ be the set of positive integers. Find all unbounded functions $f : \mathbb{N} \to \mathbb{N}$ such that
\[f(n + f(m) - 1) \mid f(n) + m - 1\]for all $m, n \in \mathbb{N}$.
5 replies
Iveela
Jun 8, 2025
cursed_tangent1434
3 hours ago
Can you Tai(y)lor it?
SatisfiedMagma   1
N 3 hours ago by Alphaamss
Show that
\[\cos(x) \le \frac{\sin^3(x)}{x^3} \]for $x \in (0,\pi/2)$,
1 reply
SatisfiedMagma
Today at 4:34 AM
Alphaamss
3 hours ago
On coefficients of a polynomial over a finite field
Ciobi_   1
N Today at 1:00 AM by AndreiVila
Source: Romania NMO 2025 12.4
Let $p$ be an odd prime number, and $k$ be an odd number not divisible by $p$. Consider a field $K$ be a field with $kp+1$ elements, and $A = \{x_1,x_2, \dots, x_t\}$ be the set of elements of $K^*$, whose order is not $k$ in the multiplicative group $(K^*,\cdot)$. Prove that the polynomial $P(X)=(X+x_1)(X+x_2)\dots(X+x_t)$ has at least $p$ coefficients equal to $1$.
1 reply
Ciobi_
Apr 2, 2025
AndreiVila
Today at 1:00 AM
Collatz Conjecture
Cpw945   7
N Yesterday at 11:55 PM by maromex
Source: "Collatz Conjecture Confirmed Through Connectivity of Odd and 8mod12 Positive Integers" on viXra
Hello everyone! I am a college student who has created a potential proof for the Collatz Conjecture, which I have posted on Vixra, under the title “Collatz Conjecture Confirmed by Connectivity of Odds and 8mod12 Positive Integers”. It is in Section 2507, under the name Chloe Williams. Feel free to check it out and tell me if my solution idea would work. The link for my paper is down below.

https://vixra.org/abs/2507.0020
7 replies
Cpw945
Tuesday at 8:56 PM
maromex
Yesterday at 11:55 PM
Linear algebra problem
Feynmann123   1
N May 25, 2025 by Etkan
Let A \in \mathbb{R}^{n \times n} be a matrix such that A^2 = A and A \neq I and A \neq 0.

Problem:
a) Show that the only possible eigenvalues of A are 0 and 1.
b) What kind of matrix is A? (Hint: Think projection.)
c) Give a 2×2 example of such a matrix.
1 reply
Feynmann123
May 25, 2025
Etkan
May 25, 2025
Linear algebra problem
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Feynmann123
4 posts
#1
Y by
Let A \in \mathbb{R}^{n \times n} be a matrix such that A^2 = A and A \neq I and A \neq 0.

Problem:
a) Show that the only possible eigenvalues of A are 0 and 1.
b) What kind of matrix is A? (Hint: Think projection.)
c) Give a 2×2 example of such a matrix.
Z K Y
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Etkan
1583 posts
#2 • 1 Y
Y by Mathzeus1024
Feynmann123 wrote:
Let A \in \mathbb{R}^{n \times n} be a matrix such that A^2 = A and A \neq I and A \neq 0.

Problem:
a) Show that the only possible eigenvalues of A are 0 and 1.
b) What kind of matrix is A? (Hint: Think projection.)
c) Give a 2×2 example of such a matrix.

a) If $\lambda \in \mathbb{C}$ is an eigenvalue of $A$ then there exists $v\in \mathbb{R}^n$ such that $v\neq 0$ and $Av=\lambda v$, so\begin{align*}A^2v & =A(Av) \\
& =A(\lambda v) \\
& =\lambda Av \\
& =\lambda (\lambda v) \\
& =\lambda ^2v.
\end{align*}Hence from $A^2=A$ we get $A^2v=Av$, and so $\lambda ^2v=\lambda v$. Since $v\neq 0$, this gives $\lambda ^2=\lambda$, and so $\lambda =0$ or $\lambda =1$.
You can also solve this using the Cayley-Hamilton Theorem, but let's keep things elementary.

b) Indeed, $A$ is a projection.

c) An example of such a $2\times 2$ matrix is $A=\begin{pmatrix}1&0\\0&0\end{pmatrix}$.
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