asiandude

The string can be divided into two different parts, the straight part and the curving part (when tangent to a circle). 

There are 6 straight parts, and each has a length of 2r. The total length is 2r * 6 = 12r.

The curving part adds up to the length of one of the circles, so its length should be \(\pi r^2\).

Together the length is \(12r+\pi r^2\).

If r is incremented by 2, we just have to replace r with (r+2). The length after incrementing is \(12(r+2)+\pi(r+2)^2\).

Simplify, the length is \(\pi r^2+12r+4 \pi r +28\).

The difference between the two lengths are \((\pi r^2+12r+4 \pi r +28) - (\pi r^2+12r)=4 \pi r+28\). And the string would be that much longer. 

We can call two adjacent terms a pair. For example, 2012 and -2002 is a pair and 32, -22 is also a pair. And the sum of the two values in each pair is 10. There are 101 pairs in totally and the sum of all the pairs should be\(101\cdot10=1010.\)

That's a circle. 

CXZ forms a triangle. And two sides of a triangle (XZ and ZC) will never be parallel, so this question is probably incorrect. 

It's a hyperbola.

Ok, the sum will equal 9 in that case. But I think there are other cases too, for example when x^2-4x+2 = 1. Can you help me think of other ones? 

sorry, it's \((x^2-4x+2)^{x^2-5x+2}=1\)

\(\frac{\sqrt2+\sqrt3}{\sqrt{10}+\sqrt{14}+\sqrt{15}+\sqrt{21}}\)

Looking at the denominator, notice that \(\sqrt{10}\) and \(\sqrt{14}\) are both divisible by \(\sqrt{2}\) and\( \sqrt{15}\), \(\sqrt{21}\) are divisible by \(\sqrt{2}\). 

We can factor those out to get \(\frac{\sqrt2+\sqrt3}{\sqrt2(\sqrt5+\sqrt7)+\sqrt{3}(\sqrt5+\sqrt7)} = \frac{\sqrt2+\sqrt3}{(\sqrt2+\sqrt{3})(\sqrt5+\sqrt7)}\). Divide top and bottom by \(\sqrt2+\sqrt3\): \(\frac{1}{\sqrt5+\sqrt7}\).

Moving the radicals to the numerator: \(\frac{1}{\sqrt5+\sqrt7} = \frac{1\cdot(\sqrt5-\sqrt7)}{(\sqrt5+\sqrt7)(\sqrt5-\sqrt7)} = \frac{\sqrt{7}-\sqrt{5}}{2}\).

So the final result is: \(\boxed{\frac{\sqrt7 - \sqrt5}{2}}\). Hope this helped 

The equation would be y = 2x-16. 

So when x = 0, y = -16.

(0, -16)