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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, 5+524 or 5×524?\sqrt{ 5 + \frac5{24} }\quad \text{ or }\quad 5 \times \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 MM from initial angle θ\theta and measure the speed at the lowest point in its swing to be vmax.v_\textrm{max}.

If we replace the wrecking ball with one of mass 2M,2M, keeping the length of the chain and the angle θ\theta at which the ball is released constant, how will vmaxv_\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?

1    12    13    14    15    16    17    18    19    110    111    112=0\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 + and + \text{ and } - to make this equation true?

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