Difference between revisions of "User:Bobthesmartypants"
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A unit circle is divided into halves. One half is divided into two halves again, and one of those is divided again, and so on. Each of these sectors of the circle is rolled up into a cone, and a circular base is capped onto each one. what is the combined surface area of all of the cones? | A unit circle is divided into halves. One half is divided into two halves again, and one of those is divided again, and so on. Each of these sectors of the circle is rolled up into a cone, and a circular base is capped onto each one. what is the combined surface area of all of the cones? | ||
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| + | '''Problem 2''' Let <math>r_1,r_2,\ldots r_{2014}</math> be rational numbers, <math>z_1,z_2,\ldots z_{2014}</math> be integers. Prove that <math>\displastyle\sum_{i=1}^{2014} z_i\cdot \text{arctan }r_i</math>, given that it is defined, can be expressed in the form <math>\text{arctan }r'</math>, where <math>r'</math> is a rational number. | ||
Revision as of 00:07, 24 February 2014
Problems
Problem 1 ![[asy] draw(Circle((0,0),1)); for (int i = 0; i < 10; ++i){ draw((0,0)--dir(360/(2^i)-90)); } [/asy]](http://latex.artofproblemsolving.com/6/8/e/68e85c9591cf0d2b18fe754976ce2ff7abe22253.png)
A unit circle is divided into halves. One half is divided into two halves again, and one of those is divided again, and so on. Each of these sectors of the circle is rolled up into a cone, and a circular base is capped onto each one. what is the combined surface area of all of the cones?
Problem 2 Let 
be rational numbers, 
be integers. Prove that $\displastyle\sum_{i=1}^{2014} z_i\cdot \text{arctan }r_i$ (Error compiling LaTeX. Unknown error_msg), given that it is defined, can be expressed in the form 
, where 
is a rational number.
