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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.

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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
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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
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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
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Geometry
shactal   0
29 minutes ago
Two intersecting circles $C_1$ and $C_2$ have a common tangent that meets $C_1$ in $P$ and $C_2$ in $Q$. The two circles intersect at $M$ and $N$ where $N$ is closer to $PQ$ than $M$ . Line $PN$ meets circle $C_2$ a second time in $R$. Prove that $MQ$ bisects angle $\widehat{PMR}$.
0 replies
shactal
29 minutes ago
0 replies
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Inspired by 2025 KMO
sqing   1
N 40 minutes ago by sqing
Source: Own
Let $ a,b,c,d $ be real numbers satisfying $ a+b+c+d=0 $ and $ a^2+b^2+c^2+d^2= 6 .$ Prove that $$ -\frac{3}{4} \leq abcd\leq\frac{9}{4}$$Let $ a,b,c,d $ be real numbers satisfying $ a+b+c+d=6 $ and $ a^2+b^2+c^2+d^2= 18 .$ Prove that $$ -\frac{9(2\sqrt{3}+3)}{4} \leq abcd\leq\frac{9(2\sqrt{3}-3)}{4}$$
1 reply
sqing
an hour ago
sqing
40 minutes ago
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RMO 2024 Q1
SomeonecoolLovesMaths   25
N an hour ago by Adywastaken
Source: RMO 2024 Q1
Let $n>1$ be a positive integer. Call a rearrangement $a_1,a_2, \cdots , a_n$ of $1,2, \cdots , n$ nice if for every $k = 2,3, \cdots , n$, we have that $a_1 + a_2 + \cdots + a_k$ is not divisible by $k$.
(a) If $n>1$ is odd, prove that there is no nice arrangement of $1,2, \cdots , n$.
(b) If $n$ is even, find a nice arrangement of $1,2, \cdots , n$.
25 replies
SomeonecoolLovesMaths
Nov 3, 2024
Adywastaken
an hour ago
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4 variables
Nguyenhuyen_AG   10
N an hour ago by Butterfly
Let $a,\,b,\,c,\,d$ are non-negative real numbers and $0 \leqslant k \leqslant \frac{2}{\sqrt{3}}.$ Prove that
$$a^2+b^2+c^2+d^2+kabcd \geqslant k+4+(k+2)(a+b+c+d-4).$$hide
10 replies
Nguyenhuyen_AG
Dec 21, 2020
Butterfly
an hour ago
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2025 KMO Inequality
Jackson0423   3
N an hour ago by sqing
Source: 2025 KMO Round 1 Problem 20

Let \(x_1, x_2, \ldots, x_6\) be real numbers satisfying
\[ x_1 + x_2 + \cdots + x_6 = 6, \]\[ x_1^2 + x_2^2 + \cdots + x_6^2 = 18. \]Find the maximum possible value of the product
\[ x_1 x_2 x_3 x_4 x_5 x_6. \]
3 replies
Jackson0423
Yesterday at 4:32 PM
sqing
an hour ago
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All divisors are one more than a perfect power
Tintarn   5
N an hour ago by Nuran2010
Source: Baltic Way 2024, Problem 16
Determine all composite positive integers $n$ such that, for each positive divisor $d$ of $n$, there are integers $k\geq 0$ and $m\geq 2$ such that $d=k^m+1$.
5 replies
Tintarn
Nov 16, 2024
Nuran2010
an hour ago
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KJMO 2001 P1
RL_parkgong_0106   1
N an hour ago by JH_K2IMO
Source: KJMO 2001
A right triangle of the following condition is given: the three side lengths are all positive integers and the length of the shortest segment is $141$. For the triangle that has the minimum area while satisfying the condition, find the lengths of the other two sides.
1 reply
RL_parkgong_0106
Jun 29, 2024
JH_K2IMO
an hour ago
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Cotangential circels
CountingSimplex   5
N 2 hours ago by rong2020
Let $ABC$ be a triangle with circumcenter $O$ and let the angle bisector of $\angle{BAC}$ intersect $BC$
at $D$. The point $M$ is such that $\angle{MCB}=90^o$ and $\angle{MAD}=90^o$. Lines $BM$ and $OA$ intersect at
the point $P$. Show that the circle centered at $P$ and passing through $A$ is tangent to segment
$BC$.
5 replies
CountingSimplex
Jun 23, 2020
rong2020
2 hours ago
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Inspired by old results
sqing   3
N 2 hours ago by sqing
Source: Own
Let $a,b,c $ be reals such that $a^2+b^2+c^2=3$ .Prove that
$$(1-a)(k-b)(1-c)+abc\ge -k$$Where $ k\geq 1.$
$$(1-a)(1-b)(1-c)+abc\ge -1$$$$(1-a)(1-b)(1-c)-abc\ge -\frac{1}{2}-\sqrt 2$$
3 replies
sqing
Today at 7:36 AM
sqing
2 hours ago
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Inspired by Philippine 2025
sqing   1
N 2 hours ago by sqing
Source: Own
Let $ a,b,c,d $ be real numbers . Prove that
$$\frac{(a-1)(b-3)(c-3)(d-1)}{ (a^2+3)(b^2+3)(c^2+3)(d^2+3)} \ge -\frac{7+4\sqrt 3}{144}$$$$\frac{(a-1)(b-2)(c-2)(d-1)}{ (a^2+3)(b^2+3)(c^2+3)(d^2+3)} \ge -\frac{11+4\sqrt 7}{432}$$


1 reply
sqing
2 hours ago
sqing
2 hours ago
J
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Group theory with unexpected NT example
AndreiVila   2
N 2 hours ago by CHOUKRI
Source: Romanian District Olympiad 2025 12.1
Let $G$ be a group and $A$ a nonempty subset of $G$. Let $AA=\{xy\mid x,y\in A\}$.
[list=a]
[*] Prove that if $G$ is finite, then $AA=A$ if and only if $|A|=|AA|$ and $e\in A$.
[*] Give an example of a group $G$ and a nonempty subset $A$ of $G$ such that $AA\neq A$, $|AA|=|A|$ and $AA$ is a proper subgroup of $G$.
[/list]
Mathematical Gazette - Robert Rogozsan
2 replies
AndreiVila
Mar 8, 2025
CHOUKRI
2 hours ago
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random variables
ORguy   2
N 2 hours ago by OGMATH
Source: traffic accidents
The time (in hours) between consecutive traffic accidents can be described by the exponential random variable X with parameter f=1/60.

Find

P(X <= 60),

P(X >120), and

P(10 < X <= 100)
2 replies
ORguy
Nov 29, 2007
OGMATH
2 hours ago
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Reducing the exponents for good
RobertRogo   1
N 4 hours ago by RobertRogo
Source: The national Algebra contest (Romania), 2025, Problem 3/Abstract Algebra (a bit generalized)
Let $A$ be a ring with unity such that for every $x \in A$ there exist $t_x, n_x \in \mathbb{N}^*$ such that $x^{t_x+n_x}=x^{n_x}$. Prove that
a) If $t_x \cdot 1 \in U(A), \forall x \in A$ then $x^{t_x+1}=x, \forall x \in A$
b) If there is an $x \in A$ such that $t_x \cdot 1 \notin U(A)$ then the result from a) may no longer hold.

Authors: Laurențiu Panaitopol, Dorel Miheț, Mihai Opincariu, me, Filip Munteanu
1 reply
1 viewing
RobertRogo
May 20, 2025
RobertRogo
4 hours ago
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Convergence of complex sequence
Rohit-2006   9
N Today at 3:41 AM by Saucitom
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,
$$z_1=z_2=\cdots=z_k=1$$.
9 replies
Rohit-2006
May 17, 2025
Saucitom
Today at 3:41 AM
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invertible matrix
jokerjoestar   2
N Mar 31, 2025 by SatisfiedMagma
Let matrix $A \in M_n(R)$ such that:$A^{2025}=2025×A$. Prove that $A-E$ is invertible
2 replies
jokerjoestar
Mar 27, 2025
SatisfiedMagma
Mar 31, 2025
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invertible matrix
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Reveal topic tags
linear algebra
matrix
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jokerjoestar
150 posts
jokerjoestar
#1 h  Mar 27, 2025, 6:50 AM
Y by
Let matrix $A \in M_n(R)$ such that:$A^{2025}=2025×A$. Prove that $A-E$ is invertible
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alexheinis
10622 posts
alexheinis
#2 h  Mar 27, 2025, 9:31 AM • 3 Y
Y by jokerjoestar, SatisfiedMagma, MihaiT
The minimal polynomial divides $x^{2025}-2025x=x(x^{2024}-2025)$ hence it is coprime to $x-1$.
Or let's do it explicitly here: if $Ax=x$ then $x=A^{2025}(x)=2025 Ax=2025 x$. Hence $x=0$.
This post has been edited 1 time. Last edited by alexheinis, Mar 27, 2025, 9:46 AM
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SatisfiedMagma
461 posts
SatisfiedMagma
#3 h  Mar 31, 2025, 10:29 AM
Y by
Here's another solution found by my batchmate Amit Prakash Jena.

Solution: Let $A - I = X$, then observe that
\begin{align*} A^{2025} &= (X+I)^{2025} \\ 2025A &= \sum_{i = 0}^{2025}X^i \binom{2025}{i} \\ 2025(X+I) &= X^{2025} + \ldots + 2025X + I \\ \cancel{2025X} + 2024I &= X\underbrace{(X^{2024} + \ldots + \binom{2025}{2}X)}_{\coloneqq B} + \cancel{2025X} \end{align*}This clearly means that $X\cdot \frac{B}{2024} = I$, proving the invertibility of $X$ since everything is a square matrix. $\blacksquare$
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