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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
Different Paths Probability
Qebehsenuef   7
N 5 hours ago by GreenKeeper
Source: OBM
A mouse initially occupies cage A and is trained to change cages by going through a tunnel whenever an alarm sounds. Each time the alarm sounds, the mouse chooses any of the tunnels adjacent to its cage with equal probability and without being affected by previous choices. What is the probability that after the alarm sounds 23 times the mouse occupies cage B?
7 replies
Qebehsenuef
Apr 28, 2025
GreenKeeper
5 hours ago
\int_{0}^{\pi/4} \frac{1}{\cos 2\phi} \cdot 2 \ln(\cot \phi) \cdot 2 \, d\phi
Martin.s   0
5 hours ago
\[
I = 2 \int_{0}^{\pi/4} \frac{1}{\cos 2\phi} \cdot 2 \ln(\cot \phi) \cdot 2 \, d\phi
\]
0 replies
Martin.s
5 hours ago
0 replies
nice integral
Martin.s   1
N 5 hours ago by aiops
$$\int_0^{\pi/2} \Bigl[
\log\bigl(\sqrt{5}-\sin\theta+1\bigr)
+\log\bigl(\sqrt{5}+\sin\theta-1\bigr)
-\log\bigl(\sqrt{5}-\sin\theta-1\bigr)
-\log\bigl(\sqrt{5}+\sin\theta+1\bigr)
\Bigr]\,d\theta.
$$
1 reply
Martin.s
5 hours ago
aiops
5 hours ago
nice ecuation
MihaiT   1
N Yesterday at 7:24 PM by Hello_Kitty
Find real values $m$ , s.t. ecuation: $x+1=me^{|x-1|}$ have 2 real solutions .
1 reply
MihaiT
Yesterday at 2:03 PM
Hello_Kitty
Yesterday at 7:24 PM
Linear algebra problem
Feynmann123   1
N Yesterday at 3:51 PM by Etkan
Let A \in \mathbb{R}^{n \times n} be a matrix such that A^2 = A and A \neq I and A \neq 0.

Problem:
a) Show that the only possible eigenvalues of A are 0 and 1.
b) What kind of matrix is A? (Hint: Think projection.)
c) Give a 2×2 example of such a matrix.
1 reply
Feynmann123
Yesterday at 9:33 AM
Etkan
Yesterday at 3:51 PM
Linear algebra
Feynmann123   6
N Yesterday at 1:09 PM by OGMATH
Hi everyone,

I was wondering whether when I tried to compute e^(2x2 matrix) and got the expansions of sinx and cosx with the method of discounting the constant junk whether it plays any significance. I am a UK student and none of this is in my School syllabus so I was just wondering…


6 replies
Feynmann123
Saturday at 6:44 PM
OGMATH
Yesterday at 1:09 PM
Local extrema of a function
MrBridges   2
N Yesterday at 11:36 AM by Mathzeus1024
Calculate the local extrema of the function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$, $(x,y)\mapsto x^4+x^5+y^6$. Are they isolated?
2 replies
MrBridges
Jun 28, 2020
Mathzeus1024
Yesterday at 11:36 AM
Integral
Martin.s   3
N Yesterday at 10:52 AM by Figaro
$$\int_0^{\pi/6}\arcsin\Bigl(\sqrt{\cos(3\psi)\cos\psi}\Bigr)\,d\psi.$$
3 replies
Martin.s
May 14, 2025
Figaro
Yesterday at 10:52 AM
Reducing the exponents for good
RobertRogo   1
N Yesterday at 9:29 AM by RobertRogo
Source: The national Algebra contest (Romania), 2025, Problem 3/Abstract Algebra (a bit generalized)
Let $A$ be a ring with unity such that for every $x \in A$ there exist $t_x, n_x \in \mathbb{N}^*$ such that $x^{t_x+n_x}=x^{n_x}$. Prove that
a) If $t_x \cdot 1 \in U(A), \forall x \in A$ then $x^{t_x+1}=x, \forall x \in A$
b) If there is an $x \in A$ such that $t_x \cdot 1 \notin U(A)$ then the result from a) may no longer hold.

Authors: Laurențiu Panaitopol, Dorel Miheț, Mihai Opincariu, me, Filip Munteanu
1 reply
RobertRogo
May 20, 2025
RobertRogo
Yesterday at 9:29 AM
Sequence divisible by infinite primes - Brazil Undergrad MO
rodamaral   5
N Yesterday at 8:01 AM by cursed_tangent1434
Source: Brazil Undergrad MO 2017 - Problem 2
Let $a$ and $b$ be fixed positive integers. Show that the set of primes that divide at least one of the terms of the sequence $a_n = a \cdot 2017^n + b \cdot 2016^n$ is infinite.
5 replies
rodamaral
Nov 1, 2017
cursed_tangent1434
Yesterday at 8:01 AM
Reduction coefficient
zolfmark   2
N Yesterday at 7:42 AM by wh0nix

find Reduction coefficient of x^10

in(1+x-x^2)^9
2 replies
zolfmark
Jul 17, 2016
wh0nix
Yesterday at 7:42 AM
a^2=3a+2imatrix 2*2
zolfmark   4
N Yesterday at 2:44 AM by RenheMiResembleRice
A
matrix 2*2

A^2=3A+2i
A^3=mA+Li


i means identity matrix,

find constant m ، L
4 replies
zolfmark
Feb 23, 2019
RenheMiResembleRice
Yesterday at 2:44 AM
Find solution of IVP
neerajbhauryal   3
N Yesterday at 12:47 AM by MathIQ.
Show that the initial value problem \[y''+by'+cy=g(t)\] with $y(t_o)=0=y'(t_o)$, where $b,c$ are constants has the form \[y(t)=\int^{t}_{t_0}K(t-s)g(s)ds\,\]

What I did
3 replies
neerajbhauryal
Sep 23, 2014
MathIQ.
Yesterday at 12:47 AM
Primes and automorphisms
CatalinBordea   2
N Saturday at 5:07 PM by a_0a
Source: Romanian District Olympiad 2016, Grade XII, Problem 3
Let be a group $ G $ of order $ 1+p, $ where $ p $ is and odd prime. Show that if $ p $ divides the number of automorphisms of $ G, $ then $ p\equiv 3\pmod 4. $
2 replies
CatalinBordea
Oct 5, 2018
a_0a
Saturday at 5:07 PM
complex integral with two circle (contour) against each other
azzam2912   4
N Apr 19, 2025 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
Apr 19, 2025
complex integral with two circle (contour) against each other
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G H BBookmark kLocked kLocked NReply
Source: seleksi onmipa itb 2022
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azzam2912
180 posts
#1
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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!
Z K Y
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alexheinis
10624 posts
#2
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The definition is $\int_{C_1} f(z) dz+\int_{C_2} f(z) dz$, each of them can be easily calculated with the residue theorem. The only pole inside $C_1$ is $z=2$, there are no poles inside $C_2$.
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azzam2912
180 posts
#3
Y by
alexheinis wrote:
The definition is $\int_{C_1} f(z) dz+\int_{C_2} f(z) dz$, each of them can be easily calculated with the residue theorem. The only pole inside $C_1$ is $z=2$, there are no poles inside $C_2$.

So the integral is added up for each contour (where each contour against each other). It means, if each contour have same orientation, the integral become minus?
for the example: if $C_1$ and $C_2$ are anticlockwise, then
$\int_{C_1} f(z) dz-\int_{C_2} f(z) dz$

is my understanding true?
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alexheinis
10624 posts
#4 • 2 Y
Y by Mango247, pokoknyaakuimut
The orientation of $C_1$ is already contained in the name $C_1$: it doesn't just mean the circle but the circle together with its orientation. Same for $C_2$. So suppose $C_1=C(0,3)$ and $C_2=C(0,1)$ both anticlockwise. Then we still have $\int_{C_1\cup C_2}=\int_{C_1}+\int_{C_2}$. Union of paths means that we run through one path and then through other the one. In each path we use the respective orientation.
Attachments:
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Mathzeus1024
921 posts
#5
Y by
The poles of $f(z) = \frac{z^4-16z^2}{z^2+3z-10} = \frac{z^4-16z^2}{(z+5)(z-2)}$ include $z = -5, 2$. We wish to compute:

$\int_{C_{1} \cup C_{2}} f(z) dz = \int_{C_{1}} f(z) dz + \int_{C_{2}} f(z) dz$

for $C_{1}: |z|=3$ & CCW; $C_{2}:|z|=1$ & CW. Since neither pole of $f$ lies within $C_{2}$, and only $z=2$ lies within $C_{1}$, our computation reduces to (via Residue Theorem):

$\oint_{|z|=3} \frac{z^4-16z^2}{z^2+3z-10} dz = 2\pi i \cdot Res_{z\rightarrow 2}f(z) = 2\pi i\left[\frac{z^4-16z^2}{2z+3}\right]_{z=2} = \textcolor{red}{-\frac{96\pi i}{7}}$.
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