# Tag Info

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 $n$...

### 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 ...
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 ...

### $E := \mathbb{Q}( \sqrt{3}, \sqrt{5}, \sqrt{7})/ \mathbb{Q}$. 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....

### 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 ...
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 ...
1 vote

### $E := \mathbb{Q}( \sqrt{3}, \sqrt{5}, \sqrt{7})/ \mathbb{Q}$. Determine a $ϑ ∈ E$ with $E = \mathbb{Q}(ϑ)$.

Let us consider the numbers $a=\sqrt 3$, $b=\sqrt 5$, $c=\sqrt7$, and $$t = (a+b)c = (\sqrt 3+\sqrt 5)\sqrt7\ .$$ We want to show that the inclusion $K:=\Bbb Q(t)\subseteq \Bbb Q(a,b,c)$ is in ...
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 ...
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 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 ...
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 $x=25$
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 ...
1 vote

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 \...
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 ...