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2018-09-10 Intermediate

         

Creed plays a carnival game where he uses a spring to launch a block of wood. If he compresses the spring by \(x\) from its natural length and releases it, the block rises to a height of \(h\) from the point of release.

If he compresses the spring down the block by \(2x\) from its natural length, how high will the block go up from the point of release?

Is there an integer \(n>1\) such that \(n!\) is a perfect square?

A rectangle of a variable size is chosen at random from the \(8 \times 5\) grid.

What is the probability the chosen rectangle has a vertex at point \({\color{red}A}?\)

You are playing a game with 2018 bins arranged in a circle. At the start, each bin has a white or black ball inside, with the two colors alternating between adjacent bins all throughout the circle.

The rules are as follows:

  • Each turn, you select two balls. Any white ball you select moves clockwise to the next bin. Any black ball you select moves counterclockwise to the next bin.
  • You may repeat this process as many times as you like.
  • The two balls selected can come from the same bin at later stages when a single bin can have multiple balls.
  • You win the game if all 2018 balls are in the same bin after a turn.

Is it possible to win this game?

A triangle’s perimeter and area have the same integer value.

What is the smallest possible area of the triangle?

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