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2018-04-02 Basic

         

A hollow cylinder is completely filled with frozen water.

Will it take longer to roll down an inclined plane while it’s frozen, or after it melts?

Assume: When frozen, the water does not move relative to the cylinder.

What is the fewest number of colors needed such that no two adjacent vertices are colored the same?

Note: In graph theory, two vertices are adjacent if they share an edge.

\[\pm1\pm2\pm3\pm \cdots \pm2017\pm2018 = \, ?\]

Using appropriate + and - signs on the left side of the equation, which number can be obtained on the right?

Triangle \(ABC\) is inscribed in a circle. The angle \(ABC\) is obtuse, and the altitude \(CH\) is tangent to the circle.

If \(HB=10\) and \(BA=8,\) what is the radius of the circle?

Tom was born on a Thursday in either 2014, 2015, 2016, or 2017. What is the oldest age Tom can turn on his next Thursday birthday?

Hint: Consider the remainder when the number of days in a year (365) is divided by the number of days in a week (7). But don't forget about leap years!

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