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2017-12-04 Advanced


2721,3721,4721,,72721 2^{72} - 1, \quad 3^{72} - 1, \quad 4^{72} - 1,\quad \ldots , \quad 72^{72} - 1

Find the greatest common divisor of the numbers above.

Two half-spheres with fixed surface charge densities σ1\sigma_1 and σ2\sigma_2 and radius RR are brought into contact, as shown to the right.

Find the force of interaction between the two half-spheres.

In a soccer match, the striker ((point S)S) has beaten all defenders and is running parallel to the sideline towards the goal line. At what distance xx from the goal line should the striker shoot at the goal for the best chance of a goal? Give the result in meters and round to one decimal place.

Details and Assumptions:

  • The striker will shoot the ball in a straight line on the ground.
  • The striker isn't perfectly accurate, so his shot may be slightly off from where he intends.

You are playing pool on a billiard table which is an equilateral triangle ABC.ABC. The ball is at the center of the triangle, and you hit it in any direction with uniform probability. What is the expected number of sides the ball hits before hitting side AB,\overline{AB}, to 3 decimal places?

Details and Assumptions:

  • The ball bounces off the sides of the table according to the law of reflection.
  • If the first side the ball hits is AB,\overline{AB}, then the number of walls the ball hits before hitting AB\overline{AB} is zero.
  • The ball is infinitesimally small.

Let SS be a subset of [0,1][0,1] consisting of a union of 1010 disjoint closed intervals I1,I2,,I10.I_1, I_2, \ldots, I_{10}.

Suppose SS has the property that for every d[0,1],d \in [0,1], there are two points x,ySx,y \in S such that xy=d.|x-y|=d.

Letting s=n=110length(In),s = \sum\limits_{n=1}^{10} \text{length}(I_n), what is the minimum possible value of s?s?

Your answer should be a rational number pq,\frac{p}{q}, where pp and qq are coprime positive integers.

Find p+q.p+q.

Bonus: Describe the sets SS for which the minimum is attained.


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