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The matrix in some degree is a scalar
FFA21   4
N Today at 12:06 PM 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
Today at 12:06 PM
Prove the statement
Butterfly   10
N Today at 10:16 AM 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
Today at 10:16 AM
Definite integration
girishpimoli   1
N Today at 9:34 AM 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
Today at 9:34 AM
Weird integral
Martin.s   0
Today at 9:33 AM
\[
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} 
\frac{1 - e^{-2} \cos\left(2\left(u + \tan u\right)\right)}
{1 - 2e^{-2} \cos\left(2\left(u + \tan u\right)\right) + e^{-4}} 
\, \mathrm{d}u
\]
0 replies
Martin.s
Today at 9:33 AM
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
May 18, 2025
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
Function on positive integers with two inputs
Assassino9931   2
N Apr 23, 2025 by Assassino9931
Source: Bulgaria Winter Competition 2025 Problem 10.4
The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$. Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$. Determine all possible values of $f(2025, 2025)$.
2 replies
Assassino9931
Jan 27, 2025
Assassino9931
Apr 23, 2025
Function on positive integers with two inputs
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Source: Bulgaria Winter Competition 2025 Problem 10.4
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Assassino9931
1361 posts
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The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$. Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$. Determine all possible values of $f(2025, 2025)$.
This post has been edited 1 time. Last edited by Assassino9931, Jan 27, 2025, 10:06 AM
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how_to_what_to
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bumpthis
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Assassino9931
1361 posts
#3
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Official Solution
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