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2018-11-19 Intermediate

         

This annulus has been formed from two concentric circles. A chord of length \(t\) of the larger circle is then drawn tangent to the inner circle.

If you only know the value of \(t,\) can you calculate the area of the annulus?

French mathematician Joseph Bertrand posed the following problem in 1889:

You have 3 identical boxes, each containing 2 coins: the first box 2 gold coins, the second box 2 silver coins, and the third box 1 gold and 1 silver coin.

Your friend shuffles the boxes at random. Then, you choose a box and pull a coin out of it, and it's gold.

What is the probability that the other coin in the same box is also a gold coin?

I have a square puzzle consisting of \( 10 \times 10 = 100 \) pieces.

If I pick two pieces at random, what is the probability that they fit together, i.e. they lie next to each other in the puzzle?

If the probability can be written as \(\frac ab\) with coprime positive integers \(a\) and \( b,\) give your answer as \(a+b.\)

Details and Assumptions:

  • Every piece is equally likely to be picked. I don't look whether the first piece has a straight edge or similar tricks.
  • Every piece only fits together with its neighboring pieces, and there is a unique solution to the puzzle.

Four congruent semicircles are packed into a square.

What proportion of the square is filled?

The sum of the ages of my five nieces is 47. Their ages are positive integers, and any two of them have a common divisor greater than 1.

How old is the eldest?

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