G
Topic
First Poster
Last Poster
Most Evil and Brutal Integral Ever Officially Proposed for an Integration Bee
Silver08 3
N
by Silver08
Source: UK University Integration Bee 2024-2025 Round 2 Relay (Singapore)
Compute: ![\[ \int_{1}^{2}e^{x( x+\sqrt{x^2-1} )}dx \]](//latex.artofproblemsolving.com/b/1/d/b1d1f5525bb9cb5fe1f1a7491ff5209cad5438d3.png)
![\[ \int_{1}^{2}e^{x( x+\sqrt{x^2-1} )}dx \]](http://latex.artofproblemsolving.com/b/1/d/b1d1f5525bb9cb5fe1f1a7491ff5209cad5438d3.png)
3 replies
Pressing 'go down button' always creates a gray box on the last post
Craftybutterfly 4
N
by Craftybutterfly
Summary of the problem: Pressing go down to last post button always creates a gray box overlapping last post
Page URL: any forum
Steps to reproduce:
1. Go to any topic in a forum
2. The gray box at the bottom overlaps part of the first post
Expected behavior: Should not show a gray box
Frequency: 100% of the time
Operating system(s): Linux HP EliteBook 835 G8 Notebook PC
Browser(s), including version: Chrome 133.0.6943.142 (Official Build) (64-bit) (cohort: Stable)
Additional information: It works on any other device, on my iPhone XR, a MacOS, and my iPad. Took the screenshot a month ago. The gray box still appears
Page URL: any forum
Steps to reproduce:
1. Go to any topic in a forum
2. The gray box at the bottom overlaps part of the first post
Expected behavior: Should not show a gray box
Frequency: 100% of the time
Operating system(s): Linux HP EliteBook 835 G8 Notebook PC
Browser(s), including version: Chrome 133.0.6943.142 (Official Build) (64-bit) (cohort: Stable)
Additional information: It works on any other device, on my iPhone XR, a MacOS, and my iPad. Took the screenshot a month ago. The gray box still appears
4 replies
IMC 2018 P4
ThE-dArK-lOrD 17
N
by sangsidhya
Source: IMC 2018 P4
Find all differentiable functions 
such that
Proposed by Orif Ibrogimov, National University of Uzbekistan
such that
Proposed by Orif Ibrogimov, National University of Uzbekistan
17 replies
f(x)=x-xe^(-1/x)
Sayan 6
N
by kamatadu
Source: ISI (BS) 2006 #6
(a) Let 
. Show that 
is an increasing function on 
, and 
.
(b) Using part (a) or otherwise, draw graphs of
, and 
for 
using the same 
and 
axes.
. Show that 
is an increasing function on 
, and 
.(b) Using part (a) or otherwise, draw graphs of
, and 
for 
using the same 
and 
axes.
6 replies
Square of a rational matrix of dimension 2
loup blanc 7
N
by ysharifi
The following exercise was posted -two months ago- on the Website StackExchange; cf.
https://math.stackexchange.com/questions/5006488/image-of-the-squaring-function-on-mathcalm-2-mathbbq
There was no solution on Stack.
-Statement of the exercise-
We consider the matrix function
.
Find the image of
.
In other words, give a method to decide whether a given matrix has or does not have at least a square root
in
; if the answer is yes, then give a method to calculate at least one of its roots.
https://math.stackexchange.com/questions/5006488/image-of-the-squaring-function-on-mathcalm-2-mathbbq
There was no solution on Stack.
-Statement of the exercise-
We consider the matrix function
.Find the image of
.In other words, give a method to decide whether a given matrix has or does not have at least a square root
in
; if the answer is yes, then give a method to calculate at least one of its roots.
7 replies
find the isomorphism
nguyenalex 10
N
by Royrik123456
I have the following exercise:
Let
be an algebraic extension of 
, and let 
be an algebraic closure of containing . Prove that if 
is an embedding such that 
for all 
, then 
extends to an automorphism of .
My attempt:
Theorem (*): Suppose that is an algebraic extension of the field , is an algebraically closed field, and 
is an embedding. Then, there exists an embedding 
that extends . Moreover, if is an algebraic closure of and is an algebraic extension of 
, then 
is an isomorphism.
Back to our main problem:
Since
and is an algebraic extension of , it follows that is an algebraic extension of . Assume that there exists an embedding such that for all . By Theorem (*), there exists an embedding 
that extends . Since is algebraically closed, 
is also an algebraically closed field.
Furthermore, because for all and is an extension of , we have
This implies that is an algebraic extension of . We conclude that 
, meaning that is an automorphism. (Finished!!)
Let choose
be the field of algebraic numbers, 
. Consider the embedding 
defined by
Then, according to the exercise above, extends to an isomorphism
How should we interpret 
?
Let
be an algebraic extension of 
, and let 
be an algebraic closure of containing . Prove that if 
is an embedding such that 
for all 
, then 
extends to an automorphism of .My attempt:
Theorem (*): Suppose that
is an algebraic extension of the field , is an algebraically closed field, and 
is an embedding. Then, there exists an embedding 
that extends . Moreover, if is an algebraic closure of and is an algebraic extension of 
, then 
is an isomorphism.Back to our main problem:
Since
and is an algebraic extension of , it follows that is an algebraic extension of . Assume that there exists an embedding such that for all . By Theorem (*), there exists an embedding 
that extends . Since is algebraically closed, 
is also an algebraically closed field.Furthermore, because
for all and is an extension of , we have
This implies that
is an algebraic extension of . We conclude that 
, meaning that is an automorphism. (Finished!!)Let choose
be the field of algebraic numbers, 
. Consider the embedding 
defined by
Then, according to the exercise above, extends to an isomorphism
How should we interpret 
?10 replies
Generating functions and recursions smelling from 1000 km
Assassino9931 12
N
by sangsidhya
Source: IMC 2022 Day 1 Problem 3
Let 
be a prime number. A flea is staying at point 
of the real line. At each minute,
the flea has three possibilities: to stay at its position, or to move by
to the left or to the right.
After
minutes, it wants to be at again. Denote by 
the number of its strategies to do this
(for example,
: it may either stay at for the entire time, or go to the left and then to the
right, or go to the right and then to the left). Find modulo .
be a prime number. A flea is staying at point 
of the real line. At each minute,the flea has three possibilities: to stay at its position, or to move by
to the left or to the right.After
minutes, it wants to be at again. Denote by 
the number of its strategies to do this(for example,
: it may either stay at for the entire time, or go to the left and then to theright, or go to the right and then to the left). Find
modulo .
12 replies
ISI 2018 #3
integrated_JRC 34
N
by anudeep
Source: ISI 2018 B.Stat / B.Math Entrance Exam
Let 
be a continuous function such that for all 
and for all 
, 
Show that is a constant function.
be a continuous function such that for all 
and for all 
, 
Show that is a constant function.
34 replies
Integration Bee Kaizo
Calcul8er 40
N
by Figaro
Hey integration fans. I decided to collate some of my favourite and most evil integrals I've written into one big integration bee problem set. I've been entering integration bees since 2017 and I've been really getting hands on with the writing side of things over the last couple of years. I hope you'll enjoy!
40 replies
A polynomial problem which originates from a combinatorical problem
BlueCloud12 0
- Suppose 
is irrational, and 
and 
are increasing positive integer sequences such that 
and ![$\beta_n = [t \alpha_n]$](//latex.artofproblemsolving.com/e/e/4/ee4b43b2d08e446fba1c4dc8d35d1ff93b10ed75.png)
. Prove that:
- (1) There are exactly
roots of 
in 
;
- (2)Denote these roots as
. Then 
converges.
is irrational, and 
and 
are increasing positive integer sequences such that 
and ![$\beta_n = [t \alpha_n]$](http://latex.artofproblemsolving.com/e/e/4/ee4b43b2d08e446fba1c4dc8d35d1ff93b10ed75.png)
. Prove that:- (1) There are exactly
roots of 
in 
;- (2)Denote these roots as
. Then 
converges.
0 replies
New Forums Duplicates
k1glaucus 3
N
by SpeedCuber7
Summary of the problem: In the New Forums collection, many of the forums are duplicated, although some have slightly different information (likely due to the forum being edited by its admin(s)). Typically only one of the two has threads, and I believe this is the only way to access
Page URL: https://artofproblemsolving.com/community/c74_new_forums
Steps to reproduce:
1. Go to the link
2. You should see duplicates of some forums
Expected behavior: Each forum appears once
Frequency: Every time
Additional information: Typically only one of the two has threads, and from creating forums in the past I believe this is the only way to access the duplicate. Also not all of the forums are duplicated. Another reason to suspect these are duplicates are the forums with similar names have the same admin.
Page URL: https://artofproblemsolving.com/community/c74_new_forums
Steps to reproduce:
1. Go to the link
2. You should see duplicates of some forums
Expected behavior: Each forum appears once
Frequency: Every time
Additional information: Typically only one of the two has threads, and from creating forums in the past I believe this is the only way to access the duplicate. Also not all of the forums are duplicated. Another reason to suspect these are duplicates are the forums with similar names have the same admin.
3 replies
