2018-08-06 Intermediate

         

True or False?

At any given point in time, there are at least two opposite points on Earth with exactly the same temperature.

Assume that temperature at a point is a continuous function of the coordinates.

\[f(x)=\frac{3x+2018}{2x+2017}\]

How many integers \(x\) are there such that \(f(x)\) is an integer?

Charlie didn't like the outcome last time, so Tracy suggested playing a different version of the coin-moving game. This version is played on a \(5 \times 5\) grid with a green square in the middle and a coin in the lower right corner. The rules are as follows:

  • Charlie goes first, then Charlie and Tracy take turns moving the coin.
  • The coin can only be moved one space up, left, or diagonally up-and-left.
  • If a player moves the coin to the green square, that player must move it again.
  • The first player to move the coin to the upper left corner wins.

If both play optimally, who will always win?

Last week, we saw how a cube can be sliced such that the cut is a regular hexagon.

Is it possible to make a cut that is a regular pentagon?

Two segments parallel to the sides of a parallelogram are drawn through a point on the parallelogram's diagonal.

The area of the entire figure is 2009, and the areas of the red, blue, and green sections are distinct integers which form a geometric progression in that order.

What is the maximum possible area of the red section?

Note: The diagram is not drawn to scale.

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