Difference between revisions of "User:Bobthesmartypants"
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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. | '''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. | ||
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| + | '''Problem 3''' A unit cube has square pyramids with base side length <math>1</math> and height <math>\dfrac{1}{2}</math> attached to each face such that the cube face and the base of the pyramid coincide. The vertex of each square pyramid is connected to each other to form a octahedron. What is the area of this octahedron? | ||
Revision as of 00:13, 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.
Problem 3 A unit cube has square pyramids with base side length 
and height 
attached to each face such that the cube face and the base of the pyramid coincide. The vertex of each square pyramid is connected to each other to form a octahedron. What is the area of this octahedron?
