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Sorry, I thought I had a solution but it was wrong.

Today I learn that the golden ratio has the following interesting property:

http://mathhelpforum.com/algebra/246829-expression.html#post864349

**Nehushtan**- Replies: 4

Write the equation as

and use the substitution

Just solved it!

In solving algebraically for you need to take the minus sign in front of the square root:

This value of will work. In other words the bullet hits the plane on the way up rather than on the way down.

The computation is straightforward but tricky. You just need to take extra care not to make any careless mistake.

The time taken for the bullet to rise to height is given by

I think the idea is to solve for algebraically and show that (horizontal distance travelled by bullet) (distance travelled by plane after passing gun).

It’s a unit circle so its radius is 1.

Let the speed (in kph) of the train relative to the first person be ; then its speed relative to the second person is .

Therefore:

Hence:

So the train’s length is 0.05 km – i.e. 50 m.

We have

Germany 1-0 Argentina after extra time. Well done Germany!

**Nehushtan**- Replies: 14

Germany – out of this world!

Brazil – out of this World Cup …

That’s all you can do. The quadratic polynomial

is irreducible over and .

In a way you could write

but this is probably not what you want.

Agnishom wrote:

The series is undefined as the first term is .
Agnishom wrote:

If a,b,c > 0

and abc=1Prove that a^2 + b^2 + c^2 ≥ a+b+c

Jensen’s inequality states that if is a convex function then

The function is convex; therefore

But by AM–GM.

Hence