Htam::Treats

A finite set (with a small cardinality) of fun math/science problems:
{

Note: Some of these are actually quite boring, they are aimed at your baby siblings. I have more interesting problems but I am too lazy to type them up just now.

[07.03] Bizarre math:

0 = 0 + 0 + 0 + ...
  = (1-1) + (1-1) + (1-1) + ...
  = 1 - 1 + 1 - 1 + 1 - 1 + ...
  = 1 + (-1+1) + (-1+1) + ...
  = 1 + 0 + 0 + ... = 1

(What's wrong with this?)

[06.07] Simple math: Compute the area of the polygon whose vertices are the solutions in the complex plane to x⁷ + x⁶ + ... + x + 1 = 0.
[05.23] Calculus: Find the area of the region bounded by the hypocycloid with vector equation r(t) = cos³ t i + sin³ t j.
[05.16] Physics: Suppose a resistor R lies along each edge of a cube (12 resistors in all) with connections at the vertices. Find the equivalent resistance between two diagonally opposite corners of the cube.
[05.11] Prove the volume of the ellipsoid x^2/a^2 + y^2/b^2 + z^2/c^2 = 1 is 4/3 pi abc.
[05.01] Find, with proof, the surface area and volume of a torus obtained by rotating about the z-axis the circle in the xz-plane with center (b,0,0) and radius a<b.
[04.19] Physics: A uniform hoop of rope with mass M and radius R spins with tangential speed V. What is the wave speed on the hoop of rope? (There is no gravitational field)
[04.10] Geometry: Given a point P in the interior of an equilateral triangle ABC. PA = 6, PB = 8, PC = 10. Find area ABC.

}

A finite set (with a small cardinality) of requirements for a treat:
{

You worked out the most recent problem on the list;
you did not get direct help from others concerning the solution of the problem;
you have not received a treat from me through this vehicle;
you do not have a BS in math yet;
you are not the contributor of the problem;
we personally know each other reasonably well; and
there exists a sufficiently large temporal interval T in the near future such that the norm of the difference of our respective position vectors on that interval is strictly less than that of the standard verbal communications and social interactions range (i.e. we can meet together some time).

}
If you qualify, contact me.