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# Problems of the Week

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# 2017-07-24 Intermediate

A kitten is inside each of two boxes, initially held at rest, as shown in the diagram. Kitty together with its box weighs $$\SI[per-mode=symbol]{1}{\kilo\gram},$$ and Petty together with its box weighs $$\SI[per-mode=symbol]{1.5}{\kilo\gram}$$.

When the boxes are released, which kitten will move down?

Notes

• Assume that the pulleys and strings are both massless and frictionless.

• The brown line at the bottom of the diagram shows that the other end of Kitty's rope is attached to the ground.

• When the boxes are released, they are simply let go. No ropes are cut!

The answer to the following question is an integer, which can also be expressed as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are positive integers.

What is $$2a^2-14ab+7?$$

Hint: If you aren't familiar with self-referential puzzles, you can try this similar problem.

A circle is inscribed in a hexagon, as shown in the diagram.

Is it possible that the side lengths of the hexagon are $7,9,11,13,15,17$ in some order?

The dartboard in the diagram consists of an infinite number of concentric circles. Each successively smaller circle has $$\frac{3}{4}$$ the radius of the preceding, larger circle.

A dart is thrown somewhere on the dartboard (striking uniformly at random over the entire area of the dartboard).

To 2 decimal places, what is the probability it strikes black? Assume the point of the dart is one-dimensional.

\begin{align} a^2+b+c \\ b^2+c+a \\ c^2+a+b \end{align}

If $$a, b, c$$ are positive integers, then is it possible that all three of the numbers above are perfect squares?

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