3
votes
Accepted
IMO 1987/P1 - Combinatoric approach
Your claim that $p_n(k)=\binom nk(n-k-1)!$ is not correct. In the case where $k=0$, your formula would imply that there are $(n-1)!$ permutations with zero fixed points, so $(n-1)!$ derangements of $n$...
- 72.4k
3
votes
Is there any way to solve for $k$, given $\beta \sin (k-k N)-\sin (k N+k)=0$?
For the case where $N$ is an integer
If you use what @TedShifrin proposed in comments, that is to say
$$\tan(kN)=a\,\tan (k)\qquad \text{with} \qquad a=\frac{\beta+1}{\beta-1}$$ let $x=\tan(k)$ and ...
- 256k
3
votes
Accepted
define $a@b=\frac{b^2+3a}{a+33b},$ calculate (1@2@...@100)×3303
The operation $@$ is mostly terrible with no pattern to it. However, if we rewrite it as $$a \mathbin{@} b = 3 \cdot \frac{3a+b^2}{3a+99b}$$ then it is easier to find a small spot of order in the ...
- 138k
2
votes
$E := \mathbb{Q}( \sqrt{3}, \sqrt{5}, \sqrt[3]{7})/ \mathbb{Q}$. Determine a $ϑ ∈ E$ with $E = \mathbb{Q}(ϑ)$.
The existing answer is great. I'd like to add that there is a general theorem, which you surely know, but also that this theorem's proof provides us with an algorithm for finding the primitive element....
- 35.3k
2
votes
Probability inference question
Look at the finite case with $N = 2n$ balls, of which there are $n$ balls of each color in Urn 1. The number of white balls $W$ successfully transferred to the second urn is a binomial random ...
- 129k
1
vote
Accepted
How can one find the value of x such that $\left(1-\sqrt{1-(R-1)^2}\right)-\left(1-R^{\frac{1}{2-x}}\right)^{2-x}=0$, $(0<R<1)$?
Consider that you look for the zero of function
$$f(x)=\left(1-\sqrt{1-(R-1)^2}\right)-\left(1-R^{\frac{1}{2-x}}\right)^{2-x}$$
Computing
$$f(0) >0 \qquad \text{and}\qquad f'(0)<0 $$
The first ...
- 256k
1
vote
$E := \mathbb{Q}( \sqrt{3}, \sqrt{5}, \sqrt[3]{7})/ \mathbb{Q}$. Determine a $ϑ ∈ E$ with $E = \mathbb{Q}(ϑ)$.
Let us consider the numbers $a=\sqrt 3$, $b=\sqrt 5$, $c=\sqrt[3]7$, and
$$
t = (a+b)c = (\sqrt 3+\sqrt 5)\sqrt[3]7\ .
$$
We want to show that the inclusion
$K:=\Bbb Q(t)\subseteq \Bbb Q(a,b,c)$ is in ...
- 31.5k
1
vote
Probability of one correct answer among 10 questions
Since you haven't studied the binomial distribution, one sequence of Right/Wrong answers she could have given is $WWWRWWWWWW$ with $Pr = 0.5^{10}$
But the right answer could be any of the ten, so ...
- 40k
1
vote
Probability of one correct answer among 10 questions
The binomial formula can be used, but isn't necessary in this case.
Remember that if every single outcome is equally likely in a finite sample space $S$, any given collection $E$ of events has the ...
1
vote
Probability of one correct answer among 10 questions
I believe that you are supposed to count the possible number of outcomes for the "rightness/wrongness" of the answers, using the counting techniques from section 6.1, and then count the ...
- 1,087
1
vote
Accepted
Solving 5 equations with 5 variable using Newton Raphson method
First of all, let me point out that your question is not very attractive for the MSE community - you should explain what you tried and be more specific about the difficulties you encountered (this ...
- 231
1
vote
Geometry problem where rectangle ABCD provides area of small rectangles
We have $c+x+d=(ABCD)/2$ so $(b+2)+(3+a+20) = (ABCD)/2$ also.
On the other hand, $a+b+x=(ABCD)/2$.
Then $a+b+25 = (ABCD)/2 = a+b+x$, so $x=25$
- 17.2k
1
vote
Accepted
Understand a FM question about a bond with varying interest rate.
The notation $B(t,T)$ means the price at time $t < T$ of a zero-coupon bond that is redeemable for the face value (principal) of 1€ at maturity time $T$. The buyer of this pure-discount bond is ...
- 89.7k
1
vote
Math olympiad : combinatorics problem about inequality
Hints towards a solution. If you're stuck, show what you've tried and explain why you couldn't push through.
Suppose we set $a_i$ in increasing order, and require that $ a_{i+1} - a_i \geq 1 + 3 \...
- 66.9k
1
vote
Accepted
Solving Logarithmic Expression
Start using$$x \log(x) + (1-x) \log(1-x)=-\log(2)+\sum_{n=1}^\infty \frac{2^{2 n-1}}{n (2 n-1)}\left(x-\frac{1}{2}\right)^{2 n}$$
To give an idea of the quality of the approximation, consider the ...
- 256k
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