Interesting algebra

Art of Problem Solving

AoPS Online

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

High School Olympiads Regional, national, and international math olympiads

Regional, national, and international math olympiads

3 M G

BBookmark VNew Topic kLocked

No tags match your search

M

Interesting algebra

G H J

Reveal topic tags

L

G H BBookmark kLocked kLocked NReply

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

310 posts

#1 h Apr 5, 2021, 11:18 AM • 2 Y

Y by oolite, David-Vieta

Let $n$ be a positive integer.
Let matrix $A$ be
$\begin{pmatrix} a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\ a_{2,1}&a_{2,2}& \cdots &a_{2,n}\\ \cdots &\cdots&\cdots&\cdots\\ a_{n,1}&a_{n,2}&\cdots&a_{n,n} \end{pmatrix}$ , where $\begin{cases}a_{i,j}>0\\a_{i,j}a_{j,i}=1\end{cases}(1\le i,j \le n).$
Let $c_i$ be the sum of all the elements in the $i$ -th row of $A$ , which means that $c_i=\sum\limits_{j=1}^n a_{i,j}.$
Find the maximum value of $\sum\limits_{i=1}^n \frac{1}{c_i}.$

Z K Y

The post below has been deleted. Click to close.

This post has been deleted. Click here to see post.

342 posts

oolite

#2 h Apr 5, 2021, 2:44 PM • 1 Y

Y by Mango247

Bumping this because it really does look interesting. The max is 1, but why?

It seems that matrices attaining the maximum can be quite nasty-looking, like this one (quoted to 3 d.p.) for example:

1.      0.899   2.394   2.377   2.275  14.544
1.113   1.      2.665   2.646   2.533  16.188
0.418   0.375   1.      0.993   0.95    6.075
0.421   0.378   1.007   1.      0.957   6.118
0.439   0.395   1.052   1.045   1.      6.392
0.069   0.062   0.165   0.163   0.156   1.

Edit: not so nasty after all. Experimentally, max-attaining matrices are those for which A_{i,j} = q_i / q_j for any vector q with non-zero entries. I don't know how to prove optimality, though.

This post has been edited 1 time. Last edited by oolite, Apr 5, 2021, 3:39 PM
Reason: oh, that's not such a nasty matrix after all ...

Z K Y

N Quick Reply

G

H

=

© 2025 AoPS Incorporated

a