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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
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Putnam 2014 A3
Kent Merryfield   16
N an hour ago by maromex
Let $a_0=5/2$ and $a_k=a_{k-1}^2-2$ for $k\ge 1.$ Compute \[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\] in closed form.
16 replies
Kent Merryfield
Dec 7, 2014
maromex
an hour ago
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Trigo or Complex no.?
hzbrl   1
N an hour ago by hzbrl
(a) Let $y=\cos \phi+\cos 2 \phi$, where $\phi=\frac{2 \pi}{5}$. Verify by direct substitution that $y$ satisfies the quadratic equation $2 y^2=3 y+2$ and deduce that the value of $y$ is $-\frac{1}{2}$.
(b) Let $\theta=\frac{2 \pi}{17}$. Show that $\sum_{k=0}^{16} \cos k \theta=0$
(c) If $z=\cos \theta+\cos 2 \theta+\cos 4 \theta+\cos 8 \theta$, show that the value of $z$ is $-(1-\sqrt{17}) / 4$.



I could solve (a) and (b). Can anyone help me with the 3rd part please?
1 reply
hzbrl
Yesterday at 3:49 AM
hzbrl
an hour ago
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Looking for someone to work with
midacer   3
N 3 hours ago by midacer
I’m looking for a motivated study partner (or small group) to collaborate on college-level competition math problems, particularly from contests like the Putnam, IMO Shortlist, IMC, and similar. My goal is to improve problem-solving skills, explore advanced topics (e.g., combinatorics, NT, analysis), and prepare for upcoming competitions. I’m new to contests but have a strong general math background(CPGE in Morocco). If interested, reply here or DM me to discuss
3 replies
midacer
Yesterday at 8:22 PM
midacer
3 hours ago
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Possible values of determinant of 0-1 matrices
mathematics2004   3
N Yesterday at 7:40 PM by Isolemma
Source: 2021 Simon Marais, A3
Let $\mathcal{M}$ be the set of all $2021 \times 2021$ matrices with at most two entries in each row equal to $1$ and all other entries equal to $0$.
Determine the size of the set $\{ \det A : A \in M \}$.
Here $\det A$ denotes the determinant of the matrix $A$.
3 replies
mathematics2004
Nov 2, 2021
Isolemma
Yesterday at 7:40 PM
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Matrix Row and column relation.
Schro   6
N May 1, 2025 by Schro
If ith row of a matrix A is dependent,Then ith column of A is also dependent and vice versa .

Am i correct...
6 replies
Schro
Apr 28, 2025
Schro
May 1, 2025
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Matrix Row and column relation.
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Reveal topic tags
linear algebra
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Schro
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Schro
#1 h  Apr 28, 2025, 2:54 PM
Y by
If ith row of a matrix A is dependent,Then ith column of A is also dependent and vice versa .

Am i correct...
Z K Y
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Samujjal101
2801 posts
Samujjal101
#2 h  Apr 28, 2025, 3:06 PM
Y by
What do you mean by dependent ?
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rchokler
2975 posts
rchokler
#3 h  Apr 28, 2025, 8:40 PM
Y by
For square matrices, if the rows are linearly dependent, then so are the columns and vice-versa. This is because for every matrix of any size, the row rank and column rank are equal.
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Schro
4 posts
Schro
#4 h  Apr 29, 2025, 4:31 PM
Y by
Samujjal101 wrote:
What do you mean by dependent ?


I mean if ith row is a combination of other rows.....then ith column will also be a combination of other columns
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Schro
4 posts
Schro
#5 h  Apr 29, 2025, 4:32 PM
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rchokler wrote:
For square matrices, if the rows are linearly dependent, then so are the columns and vice-versa. This is because for every matrix of any size, the row rank and column rank are equal.

I m not talking in general sense....
I m being specific.....if "i th" row is a combination of other rows.....then "i th " column will be a combination of other columns and vice versa
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Etkan
1576 posts
Etkan
#6 h  Apr 29, 2025, 5:01 PM
Y by
Schro wrote:
rchokler wrote:
For square matrices, if the rows are linearly dependent, then so are the columns and vice-versa. This is because for every matrix of any size, the row rank and column rank are equal.

I m not talking in general sense....
I m being specific.....if "i th" row is a combination of other rows.....then "i th " column will be a combination of other columns and vice versa

It's not true. For example, in the matrix$$\begin{pmatrix}0&0&0\\1&0&0\\-1&0&0\end{pmatrix}$$the first row is the sum of the second and third rows, but any linear combination of the second and third columns is $0$, so it cannot be equal to the first column.
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Schro
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Schro
#7 h  May 1, 2025, 6:20 AM
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Thnks, i got it
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