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2018-01-29 Basic


A rhombus and a square have the same side length, but are not congruent. Which has the smaller area?

A pendulum clock uses the oscillations of a pendulum, a swinging weight, to keep track of time. When the temperature rises and the length of the pendulum increases, what will happen to the clock?

Details and Assumptions

  • Ignore the effects of air resistance.

As shown in the diagram, spheres A and B are hanging from the corners of the ceiling, and are connected with a string. The centers of spheres A and B are distances \(d_A\) and \(d_B\) apart from their respective walls.

What can we say about the respective masses \(M_A\) and \(M_B\) of the spheres based on the distances \(d_A\) and \(d_B?\)

Assume that the system is in equilibrium, that the string joining the masses is taut and horizontal, that the spheres are the same size, and that \(d_B>d_A.\)

I have chosen four integers \({\color{red}a}, {\color{blue}b}, {\color{green}c},\) and \({\color{purple}d}\) so that all five of the following sums are values divisible by 3:

  • \({\color{red}a}+{\color{blue}b}+{\color{green}c}\)
  • \({\color{red}a}+{\color{blue}b}+{\color{purple}d}\)
  • \({\color{red}a}+{\color{green}c}+{\color{purple}d}\)
  • \({\color{blue}b}+{\color{green}c}+{\color{purple}d}\)
  • \({\color{red}a} + {\color{black}0}+{\color{black}0}. \)

Are \({\color{blue}b},\) \({\color{green}c},\) and \({\color{purple}d}\) also guaranteed to be divisible by 3?

The 4 types of triangles below are distinct but have the same area. Which has the smallest perimeter?


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