Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...
141
votes
10answers
31k views
Multiple-choice question about the probability of a random answer to itself being correct
I found this math "problem" on the internet, and I'm wondering if it has an answer:
Question: If you choose an answer to this question at random, what is the probability that you will be correct?
...
46
votes
12answers
3k views
Is it morally right and pedagogically right to google answers to homework? [closed]
This is a soft question that I have been struggling with lately.
My professor sets tough questions for homework (around 10 per week).
The difficulty is such that if I attempt the questions entirely ...
42
votes
4answers
1k views
Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even
I have a question:
Suppose $f$ is continuous and even on $[-a,a]$, $a>0$ then prove that
$$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$
How can I ...
37
votes
7answers
2k views
The last digit of $2^{2006}$
My 13 year old son was asked this question in a maths challenge. He correctly guessed 4 on the assumption that the answer was likely to be the last digit of $2^6$. However is there a better ...
36
votes
9answers
2k views
How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
How can one prove the statement
$$\lim\limits_{x\to 0}\frac{\sin x}x=1$$
without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.
This is homework. In my ...
34
votes
0answers
2k views
Simplicial homology of real projective space by Mayer-Vietoris
Consider the $n$-sphere $S^n$ and the real projective space $\mathbb{RP}^n$. There is a universal covering map $p: S^n \to \mathbb{RP}^n$, and it's clear that it's the coequaliser of $\mathrm{id}: S^n ...
32
votes
6answers
2k views
Is $2^{218!} +1$ prime?
Prove that $2^{218!} +1$ is not prime.
I can prove that the last digit of this number is $7$, and that's all.
Thank you.
30
votes
6answers
1k views
What are some good math specific study habits?
What are/ were some of your good mathematician's study habits that you found really worked for you? I'm a CS major at a respected school and have a solid GPA... However, I definitely lack when it ...
30
votes
4answers
2k views
Finding the limit of $\frac {n}{\sqrt[n]{n!}}$
I'm trying to find
$$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$
I tried couple of methods: Stolz, Squeeze, D'Alambert
Thanks!
Edit: I can't use Stirling.
28
votes
7answers
915 views
Pi Estimation using Integers
I ran across this problem in a high school math competition:
"You must use the integers $1$ to $9$ and only addition, subtraction, multiplication, division, and exponentiation to approximate the ...
27
votes
2answers
411 views
Very curious properties of ordered partitions relating to Fibonacci numbers
I came across some interesting propositions in some calculations I did and I was wondering if someone would be so kind as to provide some explanations of these phenomenon.
We call an ordered ...
24
votes
3answers
10k views
Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$
Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$.
This question is just after the definition of differentiation and the theorem that if $f$ is finitely ...
23
votes
2answers
506 views
When is a sum of consecutive squares equal to a square?
We have the sum of squares of $n$ consecutive positive integers: $$S=(a+1)^2+(a+2)^2+ ... +(a+n)^2$$ Problem was to find the smallest $n$ such, that $S=b^2$ will be square of some positive integer. I ...
22
votes
3answers
1k views
Norms on C[0, 1] inducing the same topology as the sup norm
This is an old homework problem of mine that I was never able to solve. The solution may or may not involve the Baire category theorem, which I am terrible at applying.
Let $C[0, 1]$ denote the ...
22
votes
2answers
490 views
How to prove that $\frac{(5m)!(5n)!}{(m!)(n!)(3m+n)!(3n+m)!}$ is a natural number?
How to prove that $$\frac{(5m)! \cdot (5n)!}{m! \cdot n! \cdot (3m+n)! \cdot (3n+m)!}$$ is a natural number $\forall m,n\in\mathbb N$ , $m\geqslant 1$ and $n\geqslant 1$?
Thanks in advance.
