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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
Putnam 2017 B3
goveganddomath   35
N an hour ago by Kempu33334
Source: Putnam
Suppose that $$f(x) = \sum_{i=0}^\infty c_ix^i$$is a power series for which each coefficient $c_i$ is $0$ or $1$. Show that if $f(2/3) = 3/2$, then $f(1/2)$ must be irrational.
35 replies
goveganddomath
Dec 3, 2017
Kempu33334
an hour ago
Putnam 2003 B3
btilm305   31
N an hour ago by Kempu33334
Show that for each positive integer n, \[n!=\prod_{i=1}^n \; \text{lcm} \; \{1, 2, \ldots, \left\lfloor\frac{n}{i} \right\rfloor\}\](Here lcm denotes the least common multiple, and $\lfloor x\rfloor$ denotes the greatest integer $\le x$.)
31 replies
btilm305
Jun 23, 2011
Kempu33334
an hour ago
sequence of highly correlated rv pairs
Konigsberg   0
3 hours ago
For a given $0 < \varepsilon< 1$, construct a sequence of random variables $X_1,\dots,X_n$ such that
$$
\operatorname{Corr}(X_i,X_{i+1}) = 1-\varepsilon,\qquad 1\le i\le n-1.
$$Let
$$
f(\varepsilon)=\min\left\{\,n\in\mathbb N \mid \text{it is possible that }\operatorname{Corr}(X_1,X_n)<0\right\}.
$$Find positive constants $c_1$ and $c_2$ such that
$$
\lim_{\varepsilon\to0}\frac{f(\varepsilon)}{c_1\,\varepsilon^{c_2}}=1.
$$
0 replies
Konigsberg
3 hours ago
0 replies
Find the expected END time for the given process
superpi   1
N 6 hours ago by greenturtle3141
This problem suddenly popped up in my head. But I don't know how to deal with it.

There are N bulbs. All the bulbs' available time follows same exponential distribution with parameter lambda(Or any arbitrary distribution with mean $\mu$). We do following operations
1. First, turn on the all $N$ bulbs
2. For each $k >= 2$ bulbs goes out, append ONE NEW BULB and turn on (This step starts and finishes immediately when kth bulb goes out)
3. Repeat 2 until all the bulbs goes out

What is the expected terminate time for the above process for given $N, k, \lambda$?

Or, is there any more conditions to complete the problem?

1 reply
superpi
Today at 4:33 PM
greenturtle3141
6 hours ago
No more topics!
linear algebra
ay19bme   5
N Apr 2, 2025 by loup blanc
Does the matrix equation $X^3=mI_2$ is solvable over $M_{2}(\mathbb{Z})$ for every $m\in \mathbb{Z}$. Here $X\in M_{2}(\mathbb{Z})$, $I_2=\begin{pmatrix} 1& 0\\0 & 1\end{pmatrix}$.
5 replies
ay19bme
Apr 2, 2025
loup blanc
Apr 2, 2025
linear algebra
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ay19bme
277 posts
#1
Y by
Does the matrix equation $X^3=mI_2$ is solvable over $M_{2}(\mathbb{Z})$ for every $m\in \mathbb{Z}$. Here $X\in M_{2}(\mathbb{Z})$, $I_2=\begin{pmatrix} 1& 0\\0 & 1\end{pmatrix}$.
This post has been edited 1 time. Last edited by ay19bme, Apr 2, 2025, 7:08 AM
Reason: ........
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alexheinis
10623 posts
#2
Y by
Take $m=2$ as an example. Each eigenvalue $t$ of $X$ is a root of $x^3-2$ which is irreducible by Eisenstein. And also because it has no rational root. Also $[Q(t):Q]\le 2$ since $F_X(t)$ has degree 2. Contradiction. Hence for $m=2$ we have no solution in $M_2(Z)$. The equation is solvable iff $m$ is the cube of an integer.
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Euler-epii10
3 posts
#3
Y by
I would say the following about this:
Based on the Cayley-Hamilton formula
${{X}^{2}}-tX+d{{I}_{2}}={{O}_{2}}$ where $t=Tr(X),d=\det (X).$
This can also be written as:
${{X}^{3}}-t{{X}^{2}}+dX={{O}_{2}}.$
After backsubstitution we get:
$({{t}^{2}}-d)X=(m+td){{I}_{2}}$
Now try to draw the conclusion from here!
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Gryphos
1702 posts
#4
Y by
Maybe a more elementary solution: The equation implies
$$(\det X)^3 = \det (X^3) = \det (m I_2)=m^2.$$Hence $m^2$, and consequently also $m$, is the cube of an integer.
The same proof works in any dimension $n$ not divisible by $3$.
If the dimension is divisible by $3$, however, the equation is always solvable.
This post has been edited 1 time. Last edited by Gryphos, Apr 2, 2025, 8:52 AM
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ay19bme
277 posts
#5 • 1 Y
Y by paxtonw
Thanks..So $m$ must be the cube of some integer.
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loup blanc
3600 posts
#6
Y by
The sequel of the Gryphos' post #4.

i) If $3$ doesn't divide $n$, then $m=s^3$ and $X^3=s^3I_n$.

$\bullet$ If $s=0$ then $X^3=0$ and $X=Q^{-1}diag(0,\cdots,0,J_2,\cdots,J_2,J_3,\cdots,J_3)Q$, where $Q\in GL_n(\mathbb{Q})$.

But what are all the matrices $Q$ so that $X$ is an integer matrix ? (In general, the integer matrices $Q$ are not sufficient).

$\bullet$ If $s\not= 0$ then $X$ is diagonalizable over $\mathbb{C}$ and $spectrum(X)\subset\{s,sj,sj^2\}$, where $j=\exp(2i\pi/3)$.

Then $X=Q^{-1}diag(sI_p,s\begin{pmatrix}0&-1\\1&-1\end{pmatrix},\cdots,s\begin{pmatrix}0&-1\\1&-1\end{pmatrix})Q$.

But what are all the matrices $Q$ so that $X$ is an integer matrix ?

ii) If $3$ divides $n$ and $m\not= 0$, then $X^3=mI_{3p}$, $X$ is diagonalizable over $\mathbb{C}$ and $spectrum(X)\subset\{m^{1/3},m^{1/3}j,m^{1/3}j^2\}$.

If $m$ is not a cube, then $x^3-m$ is irreducible over $\mathbb{Q}$ and we can group the eigenvalues $3$ by $3$, that is, $p\times$ $(m^{1/3},m^{1/3}j,m^{1/3}j^2)$.

Then $X=Q^{-1}diag(U_1,\cdots,U_p)Q$, where $U_i=\begin{pmatrix}0&0&m\\1&0&0\\0&1&0\end{pmatrix}$.

But what are all the matrices $Q$ so that $X$ is an integer matrix ?
This post has been edited 3 times. Last edited by loup blanc, Apr 2, 2025, 6:20 PM
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