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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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