Art of Problem Solving
AoPS Online AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy AoPS Academy
Small live classes for advanced math
and language arts learners in grades 1-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

We have your learning goals covered with Spring and Summer courses available. Enroll today!
function
B
No tags match your search
M
G
Topic
First Poster
Last Poster
H
Gheorghe Țițeica 2025 Grade 12 P4
AndreiVila   2
N 5 hours ago by MS_asdfgzxcvb
Source: Gheorghe Țițeica 2025
Let $R$ be a ring. Let $x,y\in R$ such that $x^2=y^2=0$. Prove that if $x+y-xy$ is nilpotent, so is $xy$.

Janez Šter
2 replies
AndreiVila
Friday at 10:05 PM
MS_asdfgzxcvb
5 hours ago
J
H
AB=BA if A-nilpotent
KevinDB17   1
N Today at 9:51 AM by RobertRogo
Let A,B 2 complex n*n matrices such that AB+I=A+B+BA
If A is nilpotent prove that AB=BA
1 reply
KevinDB17
Today at 9:37 AM
RobertRogo
Today at 9:51 AM
J
H
Gheorghe Țițeica 2025 Grade 11 P2
AndreiVila   1
N Today at 9:39 AM by RobertRogo
Source: Gheorghe Țițeica 2025
Let $n\geq 2$ and $A,B\in\mathcal{M}_n(\mathbb{C})$ such that $$\{\text{rank}(A^k)\mid k\geq 1\}=\{\text{rank}(B^k)\mid k\geq 1\}.$$Prove that $\text{rank}(A^k)=\text{rank}(B^k)$ for all $k\geq 1$.

Cristi Săvescu
1 reply
AndreiVila
Friday at 9:42 PM
RobertRogo
Today at 9:39 AM
J
H
Integral
Martin.s   2
N Today at 9:14 AM by Entrepreneur
Show that
$$\int_{0}^{\infty} \log \left(\frac{1+a \sin^{2} bx}{1-a \sin^{2} bx} \right) \frac{1}{x^{2}} \ dx = \frac{\pi b}{2} \left( \sqrt{1+a} - \sqrt{1-a} \right)$$
2 replies
Martin.s
Dec 9, 2023
Entrepreneur
Today at 9:14 AM
J
H
Positiion vectors of complex numbers
MetaphysicalWukong   1
N Today at 8:55 AM by MetaphysicalWukong
I cant even understand the question. Can someone help me?

1 reply
MetaphysicalWukong
Today at 8:54 AM
MetaphysicalWukong
Today at 8:55 AM
J
H
Prove the inequality, very tight...
Butterfly   0
Today at 8:54 AM
Prove $$xe^x-x^3\sqrt{x+1}-\ln x>0.$$
0 replies
Butterfly
Today at 8:54 AM
0 replies
J
H
Gheorghe Țițeica 2025 Grade 11 P1
AndreiVila   3
N Today at 7:23 AM by Levieee
Source: Gheorghe Țițeica 2025
Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+y)=f(x+f(y))$ for all $x,y\in\mathbb{R}$.
3 replies
AndreiVila
Friday at 9:40 PM
Levieee
Today at 7:23 AM
J
H
Gheorghe Țițeica 2025 Grade 12 P2
AndreiVila   1
N Today at 5:40 AM by Alphaamss
Source: Gheorghe Țițeica 2025
Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function. Prove that $$\int_0^{\pi/2}f(\sin(2x))\sin x\, dx = \int_0^{\pi/2} f(\cos^2 x)\cos x\, dx.$$
1 reply
AndreiVila
Friday at 10:01 PM
Alphaamss
Today at 5:40 AM
J
H
something like MVT
mqoi_KOLA   8
N Today at 4:20 AM by Alphaamss
If $F$ is a continuous function on $[0,1]$ such that $F(0) = F(1)$, then there exists a $c \in (0,1)$ such that:

\[ F(c) = \frac{1}{c} \int_0^c F(x) \,dx \]
8 replies
mqoi_KOLA
Yesterday at 11:37 AM
Alphaamss
Today at 4:20 AM
J
H
Sequence, limit and number theory
KAME06   2
N Today at 3:54 AM by KAME06
Source: Ecuador National Olympiad OMEC level U 2023 P6 Day 2
A positive integers sequence is defined such that, for all $n \ge 2$, $a_{n+1}$ is the greatest prime divisor of $a_1+a_2+...+a_n$. Find:
$$\lim_{n \rightarrow \infty} \frac{a_n}{n}$$
2 replies
KAME06
Feb 6, 2025
KAME06
Today at 3:54 AM
J
a