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2017-09-11 Basic

         

The area of the large square is 100 and all of the small, blue squares are congruent.

What is the total area shaded blue?

Which is larger, \[\sqrt{ 5 + \frac5{24} }\quad \text{ or }\quad 5\sqrt{\frac5{24} }\,?\]

A wrecking ball is a heavy metal mass that's swung, from a crane, into structures to demolish them. Suppose we release a ball of mass \(M\) from initial angle \(\theta\) and measure the speed at the lowest point in its swing to be \(v_\textrm{max}.\)

If we replace the wrecking ball with one of mass \(2M,\) keeping the length of the chain and the angle \(\theta\) at which the ball is released constant, how will \(v_\textrm{max}\) be affected?

\(\)
Details and Assumptions:

  • Neglect air resistance.
  • The mass of the chain is zero.

Two players are playing a shortened version of badminton: a 6-point, 3-game match with no deuce. Specifically, in each game, the player who first scores 6 points wins. The winner of the match is the player who first wins 2 out of 3 games.

Is it possible for the loser to have accumulated more points than the winner?

\[\large 1 \; \square \; \dfrac{1}{2} \; \square \; \dfrac{1}{3}\; \square \; \dfrac{1}{4}\; \square \; \dfrac{1}{5} \; \square \; \dfrac{1}{6} \; \square \; \dfrac{1}{7} \; \square \; \dfrac{1}{8} \; \square \; \dfrac{1}{9} \; \square \; \dfrac{1}{10} \; \square \; \dfrac{1}{11} \; \square \; \dfrac{1}{12}= 0\]

Can you fill the boxes with \(+ \text{ and } -\) to make this equation true?

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