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2017-04-10 Advanced

         

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As shown above, a square can be dissected into squares of distinct integer side lengths.

Is it possible to dissect a cube into cubes of distinct integer side lengths?

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Clarification: The number in each square is the side length of that square.

A tennis ball rests directly on top of a cannon ball, which has a much larger mass, i.e. \(M_\textrm{cannonball} \gg M_\textrm{tennis}.\)

When the balls are dropped, to what height (in \(\si{\centi\meter}\)) will the bottom of the tennis ball bounce?

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Details and Assumptions:

  • The bottom of the cannon ball starts at a height of \(h = \SI{60}{\centi\meter}\) from the ground, and the cannon ball has diameter \(d=\SI{30}{\centi\meter}\).
  • All collisions are perfectly elastic, and wind resistance is ignored.

\(x\) and \(y\) are positive numbers such that \[\sqrt { xy } \left( x-y \right) =x+y.\] Find the minimum value of \(x+y\) to 2 decimal places.

Let \(S(n)\) be the sum of digits of a positive integer \(n\) (when written in base 10).

If \(S(n) = 4\), find the maximum value of \(S\big(n^4\big)\).

Three circles with equal radii are centered at \((1,0), (0,0),\) and \((0,1)\).

To 4 decimal places, what is the smallest radius of the circles such that we can pick a point within each circle--like A, B, C in the diagram--and connect them to form an equilateral triangle?

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