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2017-04-10 Advanced


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?

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. McannonballMtennis.M_\textrm{cannonball} \gg M_\textrm{tennis}.

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

Details and Assumptions:

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

xx and yy are positive numbers such that

xy(xy)=x+y.\sqrt { xy } \left( x-y \right) =x+y.

Find the minimum value of x+yx+y to 2 decimal places.

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

If S(n)=4S(n) = 4, find the maximum value of S(n4)S\big(n^4\big).

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