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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 2025
0 replies
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complex integral with two circle (contour) against each other
azzam2912   4
N an hour ago by Mathzeus1024
Source: seleksi onmipa itb 2022
Let $C_1$ be a circle $|z|=3$ with counterclockwise orientation and $C_2$ be a circle $|z|=1$ with clockwise orientation.
If $f(z)=\dfrac{z^4-16z^2}{z^2+3z-10}$, then the value of $\int_{C_1 \cup C_2} f(z) dz = \dots$

ps: i'm confused with the concept union of two contour. how i proceed? The reason behind solution is much appreciated. Thanks in advance!
4 replies
azzam2912
Jul 27, 2022
Mathzeus1024
an hour ago
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Differential equation ,asymptotic
Moubinool   1
N 2 hours ago by Mathzeus1024
f’’(t)=tf(t), f(0)=1,f’(0)=0

Find limit of $$\frac{f(t)t^{1/4}}{exp(2t^{3/2}/3)}$$when t tend $+\infty$
1 reply
Moubinool
Jul 21, 2020
Mathzeus1024
2 hours ago
J
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Jordan form and canonical base of a matrix
And1viper   3
N 2 hours ago by Suan_16
Find the Jordan form and a canonical basis of the following matrix $A$ over the field $Z_5$:
$$A = \begin{bmatrix} 2 & 1 & 2 & 0 & 0 \\ 0 & 4 & 0 & 3 & 4 \\ 0 & 0 & 2 & 1 & 2 \\ 0 & 0 & 0 & 4 & 1 \\ 0 & 0 & 0 & 0 & 2 \end{bmatrix} $$
3 replies
And1viper
Feb 26, 2023
Suan_16
2 hours ago
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Putnam 1938 B2
jhu08   3
N 3 hours ago by Mathzeus1024
Find all solutions of the differential equation $zz" - 2z'z' = 0$ which pass through the point $x=1, z=1.$
3 replies
jhu08
Aug 20, 2021
Mathzeus1024
3 hours ago
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Differential equations , Matrix theory
c00lb0y   1
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.
1 reply
c00lb0y
Apr 17, 2025
loup blanc
3 hours ago
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OMOUS-2025 (Team Competition) P7
enter16180   1
N 3 hours ago by RobertRogo
Source: Open Mathematical Olympiad for University Students (OMOUS-2025)
Let $R$ be a ring not assumed to have an identity, with the following properties:
(i) There is an element of $R$ that is not nilpotent.
(ii) If $x_{1}, \ldots, x_{2024}$ are nonzero elements of $R$, then $\sum_{j=1}^{2024} x_{j}^{2025}=0$.

Show that $R$ is a division ring, that is, the nonzero elements of R form a group under multiplication.
1 reply
enter16180
Yesterday at 12:04 PM
RobertRogo
3 hours ago
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Question
Riptide1901   1
N 4 hours ago by Mathzeus1024
Isn't their a left hand side of the graph that they're leaving out? I'm confused why they only consider $x>\sqrt{15}/2$ and not $x<-\sqrt{15}/2.$
1 reply
Riptide1901
Jan 29, 2025
Mathzeus1024
4 hours ago
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Putnam 2019 A5
hoeij   17
N Today at 5:34 AM by Ilikeminecraft
Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by $q(x) = \sum_{k=1}^{p-1} a_k x^k$ where $a_k = k^{(p-1)/2}$ mod $p$. Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$.
17 replies
hoeij
Dec 10, 2019
Ilikeminecraft
Today at 5:34 AM
J
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Projection of angle into planes
geekmath-31   0
Today at 5:14 AM
Question:
consider the angle formed by 2 half lines in the three dimensional space. Prove that the average of the projection of the angle into all of the planes is equal to the angle

The answer is in the attachments.

Please could anyone prove the answer to me in detail.
0 replies
geekmath-31
Today at 5:14 AM
0 replies
J
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Projection of angle into planes
geekmath-31   0
Today at 4:31 AM
Question:
consider the angle formed by 2 half lines in the three dimensional space. Prove that the average of the projection of the angle into all of the planes is equal to the angle

The answer is in the attachments.

Please could anyone prove the answer to me in detail.
0 replies
geekmath-31
Today at 4:31 AM
0 replies
J
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Soviet Union University Mathematical Contest
geekmath-31   0
Today at 3:40 AM
Given a n*n matrix A, prove that there exists a matrix B such that ABA = A

Solution: I have submitted the attachment

The answer is too symbol dense for me to understand the answer.
What I have undertood:

There is use of direct product in the orthogonal decomposition. The decomposition is made with kernel and some T (which the author didn't mention) but as per orthogonal decomposition it must be its orthogonal complement.

Can anyone explain the answer in much much more detail with less use of symbols ( you can also use symbols but clearly define it).

Also what is phi | T ?
0 replies
geekmath-31
Today at 3:40 AM
0 replies
J
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Dimension of a Linear Space
EthanWYX2009   0
Today at 2:50 AM
Source: 2024 May taca-10
Let \( V \) be a $10$-dimensional inner product space of column vectors, where for \( v = (v_1, v_2, \dots, v_{10})^T \) and \( w = (w_1, w_2, \dots, w_{10})^T \), the inner product of \( v \) and \( w \) is defined as \[\langle v, w \rangle = \sum_{i=1}^{10} v_i w_i.\]For \( u \in V \), define a linear transformation \( P_u \) on \( V \) as follows:
\[ P_u : V \to V, \quad x \mapsto x - \frac{2\langle x, u \rangle u}{\langle u, u \rangle} \]Given \( v, w \in V \) satisfying
\[ 0 < \langle v, w \rangle < \sqrt{\langle v, v \rangle \langle w, w \rangle} \]let \( Q = P_v \circ P_w \). Then the dimension of the linear space formed by all linear transformations \( P : V \to V \) satisfying \( P \circ Q = Q \circ P \) is $\underline{\quad\quad}.$
0 replies
EthanWYX2009
Today at 2:50 AM
0 replies
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Matrices and Determinants
Saucepan_man02   5
N Today at 1:23 AM by Saucepan_man02
Hello

Can anyone kindly share some problems/handouts on matrices & determinants (problems like Putnam 2004 A3, which are simple to state and doesnt involve heavy theory)?

Thank you..
5 replies
Saucepan_man02
Apr 4, 2025
Saucepan_man02
Today at 1:23 AM
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Putnam 1960 B1
sqrtX   4
N Yesterday at 11:26 PM by KAME06
Source: Putnam 1960
Find all integer solutions $(m,n)$ to $m^{n}=n^{m}.$
4 replies
sqrtX
Jun 18, 2022
KAME06
Yesterday at 11:26 PM
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J
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Problem on matrix algebra
Fly29   1
N Apr 4, 2025 by alexheinis
Is $\mathrm{GL}(\mathbb R^3)$ isomorphic to $\mathrm{GL}_3(\mathbb R)$?
1 reply
Fly29
Apr 4, 2025
alexheinis
Apr 4, 2025
J
Problem on matrix algebra
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Fly29
92 posts
Fly29
#1 h  Apr 4, 2025, 7:46 AM
Y by
Is $\mathrm{GL}(\mathbb R^3)$ isomorphic to $\mathrm{GL}_3(\mathbb R)$?
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alexheinis
10547 posts
alexheinis
#2 h  Apr 4, 2025, 8:21 AM
Y by
We fix a basis, for example the canonical basis $e_1,e_2,e_3$ of ${\bf R}^3$.
The group ${\rm GL}(R^3)$ consists of the invertible linear maps from $R^3$ to $R^3$, hence $3\times 3$-matrices over $R$ that are invertible.
The group ${\rm GL}_3(R)$ has the same elements. Since the group action in both actions is given by matrix multiplication, we indeed have ${\rm GL}(R^3)\cong {\rm GL}_3(R)$.
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