[Shirsendu Roy]

The product of norms-norm of product

\prod_{k=1}^{n-1}||e^\frac}2\piik}{n}-1||=||\prod_{k=1}^{n-1}(e^\frac{2\piik}{n}-1)||

Let w_k=e^\frac{2\piik}{n} for roots of unity other than 1, these are the roots of the equation z^{n-1}+z^{n-2}+...+z+1=0

so,(w_k-1) is a root of

(z+1)^{n-1}+(z+1)^{n-2}+...+(z+1)+1=0 for all k.

The product of the roots of the polynomial are by Vieta's formula equal to (-1)^{n-1}\frac{a_0}{a_n} where a_0 is the coefficient on the constant term a_n is the coefficient on the leading term

leading term's coefficient is 1, the constant term is the sum of n 1's.

product is (-1)^{n-1}n and its norm is n.

[Shirsendu Roy]

After three moves, the person has to be at less than 3 moves from the center.

here four such cases, 1 point has 9 ways to get there, two others have 3 ways of getting there and the last one has only 1 way.

For the way back there are as many possibilities, so sum of squares 9^2+3^2+3^2+1^2=100 multiplied by four because of symmetry=400 different paths.

If you consider a path is same as an other if it is taken in an other order then divide by 2 gives 200.

[Shirsendu Roy]

sin^5x+cos^3x  \leq |sin^5x+cos^3x|

\leq |sin^5x|+|cos^3x|

\leq |sin^2x|+|cos^2x|

=sin^2x+cos^2x

=1

or, sinx and cosx must be both 0 or 1. Putting that in given equation gives sinx=1 or cosx=1.

So, x=\pi/2+2n\pi or, x=2m\pi.

[Shirsendu Roy]

LHS=[a^5+b^5+{-(a+b)^5}]/5=-a^4b-2a^3b^2-2a^2b^3-ab^4

and a^3+b^3+c^3-3abc=0 gives (a^3+b^3+c^3)/3=abc

again RHS= [a^2+b^2+{-(a+b)}^2]abc/2=[a^2+b^2+ab]ab{-(a+b)}=-a^4b-2a^3b^2-2a^2b^3-ab^4

here LHS=RHS

[Shirsendu Roy]

Area between two intersecting circles given by formula used here as

=(R^2/4)cos^(-1)(1/4)+(R^2)cos^(-1)(7/8)-(R^2/8)(\sqrt{15})

 

area other than inside any of the semicircles and the red shaded part but inside the quarter circle is given as

=(R^2/2)(4-\pi)

area of the semi circle inside the quarter circle is given as (\piR^2)/8

 

Required area= area of the red shaded part=(\piR^2)/4-[(R^2/2)(4-\pi)+(\piR^2)/8+(R^2/4)cos^(-1)(1/4)+(R^2)cos^(-1)(7/8)-(R^2/8)\sqrt{15}] which is the required expression.

 

[Author]

Posts