User:Bobthesmartypants/Problems

Problems

Problem 1 $[asy] draw(Circle((0,0),1)); for (int i = 0; i < 10; ++i){ draw((0,0)--dir(360/(2^i)-90)); } [/asy]$

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 $r_1,r_2,\ldots r_{2014}$ be rational numbers, $z_1,z_2,\ldots z_{2014}$ 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 $\text{arctan }r'$ , where $r'$ is a rational number.

Problem 3 A unit cube has square pyramids with base side length $1$ and height $\dfrac{1}{2}$ 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?

Problem 4 $[asy] draw((0,0)--(-1/2,sqrt(3)/2)--(3/2,5*sqrt(3)/2)--(11/2,5*sqrt(3)/2)--(6,2*sqrt(3))--(4,0)--cycle); draw((0,0)--(-1/2,-sqrt(3)/2)--(3/2,-5*sqrt(3)/2)--(11/2,-5*sqrt(3)/2)--(6,-2*sqrt(3))--(4,0)--cycle,linetype("8 8")); draw((11/2,5*sqrt(3)/2)--(3/2,-5*sqrt(3)/2),red); label("$A$",(-1/2,sqrt(3)/2),dir(150)); label("$B$",(3/2,5*sqrt(3)/2),dir(90)); label("$C$",(11/2,5*sqrt(3)/2),dir(90)); label("$D$",(6,2*sqrt(3)),dir(0)); label("$E$",(4,0),dir(0)); label("$F$",(0,0),dir(180)); label("$A'$",(-1/2,-sqrt(3)/2),dir(210)); label("$B'$",(3/2,-5*sqrt(3)/2),dir(-90)); label("$C'$",(11/2,-5*sqrt(3)/2),dir(-90)); label("$D'$",(6,-2*sqrt(3)),dir(0)); label("$P$",intersectionpoint((0,0)--(4,0),(11/2,5*sqrt(3)/2)--(3/2,-5*sqrt(3)/2)),dir(130),red); dot((intersectionpoint((0,0)--(4,0),(11/2,5*sqrt(3)/2)--(3/2,-5*sqrt(3)/2))),red); [/asy]$

Equiangular hexagon $ABCDEF$ has $AB=BC=DE=EF=4$ and $AF=CD=1$ . The hexagon is reflected across the segment $\overline{EF}$ to form hexagon $A'B'C'D'EF$ . $\overline{B'C}$ intersects $\overline{EF}$ at point $P$ . Find $\dfrac{EP}{FP}$ .

Problem 5 Let $m,n$ be real numbers in the interval $[-2,2]$ . Variables $\alpha,\beta$ satisfy the system of equations: $\left\{\begin{array}{l}\sin\alpha+\cos\beta=m\\ \sin\beta+\cos\alpha=n\end{array}\right.$ Given that the numbers $m.n$ are picked at random, what is the probability that the system of equations has real solutions for $\alpha,\beta$ ?

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