fiora

There are 304 women and 228 men in that factory.So there are 304+228=532 workers in that factory.

odd integer +odd integer= even integer

even interger + odd interger = odd integer

1 3 5 7 9 11 13 15 all are odd integer

It doesnt matter what you choose from first two list ,when you add it up, it always give you a even number.

and in the third list, all are odd interger,so when calculate the sum of your integer that you get and the integer

that you choose in third list,you always get a odd integer.

So I think it is impossible to do this. 

Now, I want contiune my work.

In my last post ,we got cos 72 degrees =(-1+$${\sqrt{}}$$5)/4

Any interior angle of regular pentagon=(5-2)*180/5 degrees=108 degrees

cos 108 degrees =-cos（$${\frac{{\mathtt{\pi}}}{{\mathtt{2}}}}$$-72 degrees) =-cos72 degrees=(1-$${\sqrt{}}$$5)/4

In any pentagon ,every sides are equal and every diagonals are equal.

Lets say we are fing the ratio between AB and AC

According to law of cosine 

In triangle ABC,we have AB^2=AC^2+BC^2-2*AC*BC*cosC, rearrange we have

                     2*AC*BC*cosC=AC^2+BC^2-AB^2

                    and because AC and BC are the sides of regular pentagon,so

                    2*AC^2*cosC=2*AC^2-AB^2

Both side divide by 2*AC^2,then we get

                               cosC=1-AB^2/2*AC^2

                AB^2/2*AC^2=1-cosC

                AB^2/2*AC^2=1-(-cos72 degrees)

                AB^2/2*AC^2=1+cos72 degrees

               AB^2/2*AC^2=1+(-1+5)/4

$${\frac{\left({\mathtt{sqrt5}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}} = {\mathtt{1.618\: \!033\: \!988\: \!749\: \!894\: \!8}}$$ 

Now,I am going to rearrange the Alan's equation and check

sin18 degrees=s/2d

2*sin 18 degrees=s/d

1/(2*sin18 degrees)=d/s

d/s=1/(2*sin18 degrees)=1.6180339887496196

Yes，φ is the simple ratio between the size of the diagonal and the the size of the sides of any regular pentagon.

So,what?Do you know the background information of φ?Do you know understand why the φ is ratio betweeenthe diagonal and the sides?Can you prove it?DO you know how to use φ in a math problem or ideal life? Might be not.

And I want to say there is no always a shortcut to solve a problem.Every formula and shortcut is come from 

ancient peoples through thinking and hard working to figure it out,and reserve it for us.They might figured out the ratio between the diagonal and of any regular pentagon just like I did,or may be in a easier way.

That is it.

Here is a method to find out exact value of cos72 degrees.My idea is come from C.F.Gauss.

let cos(a) =2pi/5=x

sin(a)=-sin(2kPi-a) if k=1

sin(a)=-sin(2pi-a)=-sin(5a-a)=-sin4a=-2sin(2a)cos(2a)=-4sin（a)cos(a)cos(2a)

because -4sin(a)  equal 0,so divide the equation by -4sin(a)

simplify we have -1/4=cos(a)cos(2a)

                        -1/4=1/2*[cos(a+2a)+cos(a-2a)]

                        -1/2=cos3a+cos(-a）

because cos(3a)=cos(-3a)=cos(2kpi-3a),when k=1,we have cos(3a)=cos(5a-3a)=cos(2a),and cos(-a)=cos(a)

                    so -1/2=cos3a+cos(-a) $$\Rightarrow$$ -1/2=cos(2a)+cos(a)

remember,I setted up cos(a)=x at the start 

so cos(2a)=cos^2(a)-sin^2(a)=cos^2(a)-[1-cos^2(a)]=2cos^2(a)-1=2x^2-1

change the equation -1/2=cos(2a)+cos(a) to -1/2=2x^2-1+x$$\Rightarrow$$-1=4x^2+2x-2$$\Rightarrow$$0=4x^2+2x-1

ax^2+bx+c=0 a=4 b=2 c=-1 let x1 and x2 are the two soloutionof the equation

x1=[-b+(b^2-4ac)^1/2]/2a=[-2+(2^2-4*4(-1)^1/2]/(4*2)=[-2+20^(1/2)]/8=[-1+5^(1/2)]/4

x2=[-b-(b^2-4ac)^1/2]/2a=[-2-(2^2-4*4(-1)^1/2]/(4*2)=[-2-20^(1/2)]/8=[-1-5^(1/2)]/4

Beacause x=cos(a)=cos 72degrees>0,so cos 72 degrees=-1/4+5^(1/2)/4

What Is the length of the side of this pentagon?

I think that that that that that student wrote on the blackboard was wrong.