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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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0 replies
jlacosta
May 1, 2025
0 replies
Differential equations , Matrix theory
c00lb0y   3
N 3 hours ago by loup blanc
Source: RUDN MATH OLYMP 2024 problem 4
Any idea?? Diff equational system combined with Matrix theory.
Consider the equation dX/dt=X^2, where X(t) is an n×n matrix satisfying the condition detX=0. It is known that there are no solutions of this equation defined on a bounded interval, but there exist non-continuable solutions defined on unbounded intervals of the form (t ,+∞) and (−∞,t). Find n.
3 replies
c00lb0y
Apr 17, 2025
loup blanc
3 hours ago
The matrix in some degree is a scalar
FFA21   4
N 3 hours ago by FFA21
Source: MSU algebra olympiad 2025 P2
$A\in M_{3\times 3}$ invertible, for an infinite number of $k$:
$tr(A^k)=0$
Is it true that $\exists n$ such that $A^n$ is a scalar
4 replies
FFA21
Today at 12:11 AM
FFA21
3 hours ago
Prove the statement
Butterfly   10
N 5 hours ago by oty
Given an infinite sequence $\{x_n\} \subseteq  [0,1]$, there exists some constant $C$, for any $r>0$, among the sequence $x_n$ and $x_m$ could be chosen to satisfy $|n-m|\ge r $ and $|x_n-x_m|<\frac{C}{|n-m|}$.
10 replies
Butterfly
May 7, 2025
oty
5 hours ago
Definite integration
girishpimoli   1
N 6 hours ago by Mathzeus1024
If $\displaystyle g(t)=\int^{t^{2}}_{2t}\cot^{-1}\bigg|\frac{1+x}{(1+t)^2-x}\bigg|dx.$ Then $\displaystyle \frac{g(5)}{g(3)}$ is
1 reply
girishpimoli
Apr 6, 2025
Mathzeus1024
6 hours ago
Weird integral
Martin.s   0
6 hours ago
\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 
\frac{1 - e^{-2} \cos\left(2\left(u + \tan u\right)\right)}
{1 - 2e^{-2} \cos\left(2\left(u + \tan u\right)\right) + e^{-4}} 
\, \mathrm{d}u
\]
0 replies
Martin.s
6 hours ago
0 replies
hard number theory problem
danilorj   4
N Today at 9:01 AM by c00lb0y
Let \( a \) and \( b \) be positive integers. Prove that
\[
a^2 + \left\lceil \frac{4a^2}{b} \right\rceil
\]is not a perfect square.
4 replies
danilorj
May 18, 2025
c00lb0y
Today at 9:01 AM
Unsolving differential equation
Madunglecha   2
N Today at 8:44 AM by vanstraelen
For parameter t
I made a differential equation :
y"=y*(x')^2
for here, '&" is derivate and second order derivate for t
could anyone tell me what is equation between y&x?
2 replies
Madunglecha
Sunday at 3:33 PM
vanstraelen
Today at 8:44 AM
maximum dimention of non-singular subspace
FFA21   1
N Today at 8:27 AM by alexheinis
Source: MSU algebra olympiad 2025 P1
We call a linear subspace in the space of square matrices non-singular if all matrices contained in it, except for the zero one, are non-singular. Find the maximum dimension of a non-singular subspace in the space of
a) complex $n\times n$ matrices
b) real $4\times 4$ matrices
c) rational $n\times n$ matrices
1 reply
FFA21
Today at 12:02 AM
alexheinis
Today at 8:27 AM
functional equation
pratyush   4
N Today at 8:00 AM by Mathzeus1024
For the functional equation $f(x-y)=\frac{f(x)}{f(y)}$, if f ' (0)=p and f ' (5)=q, then prove f ' (-5) = q
4 replies
pratyush
Apr 4, 2014
Mathzeus1024
Today at 8:00 AM
a product that is never a square
FFA21   1
N Today at 7:21 AM by ohiorizzler1434
Source: MSU algebra olympiad 2025 P3
Show that the product $7*77*777*7777*77777...$ is never a square of an integer.
1 reply
FFA21
Today at 12:18 AM
ohiorizzler1434
Today at 7:21 AM
Convergence of complex sequence
Rohit-2006   8
N Today at 7:12 AM by ohiorizzler1434
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,
$$z_1=z_2=\cdots=z_k=1$$.
8 replies
Rohit-2006
May 17, 2025
ohiorizzler1434
Today at 7:12 AM
ignore this
Martin.s   7
N Today at 6:22 AM by Cats_on_a_computer
Source: ignore this
ignore this
7 replies
Martin.s
Jul 16, 2024
Cats_on_a_computer
Today at 6:22 AM
non-identity invariant subgroup of automorphism
FFA21   1
N Today at 2:18 AM by ysharifi
Source: MSU algebra olympiad 2025 P6
Show that an order two automorphism of a non-identity Abelian group always has a non-identity invariant cyclic subgroup
1 reply
FFA21
Today at 12:27 AM
ysharifi
Today at 2:18 AM
proper subfield of squares
FFA21   1
N Today at 2:01 AM by ysharifi
Source: MSU algebra olympiad 2025 P5
Show that if the set of all squares of elements of a field is a proper subfield, then the characteristic of the field is two.
1 reply
FFA21
Today at 12:25 AM
ysharifi
Today at 2:01 AM
Matrix problem
hef4875   3
N Apr 3, 2025 by loup blanc
The matrix \( A = (a_{ij}) \in Mat_p(\mathbb{C}) \) is defined by the conditions
\( a_{12} = a_{23} = \dots = a_{(p-1)p} = 1 \) and \( a_{ij} = 0 \) for a set of indices \( (i,j) \).
Prove that there do not exist nonzero matrices \( B, C \in Mat_p(\mathbb{C}) \) satisfying the equation
\[
(I_p + A)^n = B^n + C^n.
\]$\forall$ $n$ is a postive integer.
3 replies
hef4875
Mar 26, 2025
loup blanc
Apr 3, 2025
Matrix problem
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hef4875
132 posts
#1
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The matrix \( A = (a_{ij}) \in Mat_p(\mathbb{C}) \) is defined by the conditions
\( a_{12} = a_{23} = \dots = a_{(p-1)p} = 1 \) and \( a_{ij} = 0 \) for a set of indices \( (i,j) \).
Prove that there do not exist nonzero matrices \( B, C \in Mat_p(\mathbb{C}) \) satisfying the equation
\[
(I_p + A)^n = B^n + C^n.
\]$\forall$ $n$ is a postive integer.
This post has been edited 2 times. Last edited by hef4875, Mar 26, 2025, 9:51 AM
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Filipjack
873 posts
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It's this problem.
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loup blanc
3599 posts
#5 • 1 Y
Y by MS_asdfgzxcvb
Thanks @ Filipjack.
$\textbf{Proposition.}$ If $I+A=B+C$ and $(I+A)^2=B^2+C^2$, then $B=I+A,C=0$ or $B=0,C=I+A$.
$\textbf{Proof.}$ $I+A=B+C$, $(I+A)^2=B^2+C^2=B^2+C^2+BC+CB$;
then $BC+CB=0$. Since $C=I+A-B$, we deduce that
(*) $2B-2B^2+AB+BA=0$.
(*) is a Riccati equation; we associate the $2p\times 2p$ pseudo-Hamiltonian
$M=\begin{pmatrix}-A&2I_p\\0_p&2I_p+A\end{pmatrix}\in M_{2p}(\mathbb{C})$. There is a one to one correspondence between
the set of solutions of (*) and the set of $p$-dimensional $M$-invariant subspaces $E$ s.t.
(1) $E\oplus span(e_{p+1},\cdots,e_{2p})=span(e_1,\cdots,e_{2p})$.
The Jordan form of $M$ -in the new basis $(f_1,\cdots,f_{2p})$- is $diag(J_p[2],J_p[0])$, where
$J_p[\lambda]=\lambda I_p+J_p$ and $J_p$ is the nilpotent Jordan block of dimension $p$.
$E$ is in one of those $p+1$ forms $span(f_1,\cdots f_q,f_{p+1},\cdots,f_{2p-q}),q=0,\cdots,p$.
Only the following $2$ spaces verify condition (1):
$\ker(M^p)=span(e_1,\cdots,e_p)$ and $\ker((M-2I_p)^p)$.
$\bullet$ It's complicated to write a proof of the previous line and I don't have any nice proof.
Thus (*) has $2$ solutions. Luckily, we have two obvious solutions: $B=0$ and
$B=I+A$. $\square$
This post has been edited 3 times. Last edited by loup blanc, Mar 27, 2025, 8:32 PM
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loup blanc
3599 posts
#10
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Here is a complete resolution of the above equation (*).
We put $A=J$, the nilpotent Jordan block of dimension $n$.
Let $U=I+J-B,V=B$; then $U+V=I+J,UV=-VU$.
$(U+V)^2=(U-V)^2=U^2+V^2=I+2J+J^2$. Thus $U+V,U-V$ commute with $2J+J^2$, then with $J$.
Therefore $U+V,U-V$ are polynomials in $J$ and $U,V$ too. In particular $UV=VU=0$.
Finally $B$ is a polynomial in $J$, $B=\sum_{i=0}^n b_iJ^i$.
Case 1. $b_0\not= 1$. Then $U$ is invertible and $V=B=0$.
Case 2. $b_0= 1$. Then $V=B$ is invertible and $U=0$, that is, $B=I+J$. $\square$
This post has been edited 2 times. Last edited by loup blanc, Apr 3, 2025, 3:47 PM
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