\[ 2^{72} - 1, \quad 3^{72} - 1, \quad 4^{72} - 1,\quad \ldots , \quad 72^{72} - 1 \]
Find the greatest common divisor of the numbers above.
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Two half-spheres with fixed surface charge densities \(\sigma_1\) and \(\sigma_2\) and radius \(R\) are brought into contact, as shown to the right.
Find the force of interaction between the two half-spheres.
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In a soccer match, the striker \((\)point \(S)\) has beaten all defenders and is running parallel to the sideline towards the goal line. At what distance \(x\) 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.
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You are playing pool on a billiard table which is an equilateral triangle \(ABC.\) 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 \(\overline{AB},\) to 3 decimal places?
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Let \(S\) be a subset of \([0,1]\) consisting of a union of \(10\) disjoint closed intervals \(I_1, I_2, \ldots, I_{10}.\)
Suppose \(S\) has the property that for every \(d \in [0,1],\) there are two points \(x,y \in S\) such that \(|x-y|=d.\)
Letting \(s = \sum\limits_{n=1}^{10} \text{length}(I_n),\) what is the minimum possible value of \(s?\)
Your answer should be a rational number \(\frac{p}{q},\) where \(p\) and \(q\) are coprime positive integers.
Find \(p+q.\)
Bonus: Describe the sets \(S\) for which the minimum is attained.
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