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2018-06-04 Basic


A sheet of paper has width 21.0 cm21.0\text{ cm} and length 29.7 cm.29.7\text{ cm}. When rolled up, it can form the lateral surface of two different cylinders:

  • A long, slim cylinder, using the length as its height.
  • A short, fat cylinder, using the width as its height.

Which cylinder has the larger volume?

What is the length of the diagonal?

Professional sports leagues usually do their best to even the playing field by distributing skilled players throughout the league. This way, teams are always competitive and the games are more fun to watch.

Still, teams are always a bit uneven in terms of skill level, or scoring potential. Throughout the 2018 NBA season, the Cleveland Cavaliers and the Golden State Warriors have scored an average of 101 and 109 points per game, respectively and are facing each-other in the NBA Championship. In the NHL, the Washington Capitals and the Las Vegas Knights scored an average of 3.12 and 3.26 goals per game, respectively and are facing each-other in the Stanley Cup.

An example basketball scoreboard. An example basketball scoreboard.

Which underdog team is more likely to win their respective championship?

Details: Assume the teams' defensive performance is comparable and use only the provided points-per-game statistics. In hockey, a goal counts as one point; in basketball, goals count as either one, two, or three points depending on how it was made.

All of the digits from 0 through 9 are used to create two 5-digit positive integers, and then the two numbers are subtracted to find their positive difference: 7291054836=18074.72910 - 54836 = {\color{#D61F06}18074}. What's the smallest positive difference we can get this way?

Two unit circles are joined by many equally spaced strings of length 22 to form a cylinder. The upper circle is rotated by 9090^\circ while the lower is fixed. Naturally, the two circles come closer together.

What is the new distance between them?


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