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# Problems of the Week

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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?


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?


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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