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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 dAd_A and dBd_B apart from their respective walls.

What can we say about the respective masses MAM_A and MBM_B of the spheres based on the distances dAd_A and dB?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 dB>dA.d_B>d_A.

I have chosen four integers a,b,c,{\color{#D61F06}a}, {\color{#3D99F6}b}, {\color{#20A900}c}, and d{\color{#69047E}d} so that all five of the following sums are values divisible by 3:

  • a+b+c{\color{#D61F06}a}+{\color{#3D99F6}b}+{\color{#20A900}c}
  • a+b+d{\color{#D61F06}a}+{\color{#3D99F6}b}+{\color{#69047E}d}
  • a+c+d{\color{#D61F06}a}+{\color{#20A900}c}+{\color{#69047E}d}
  • b+c+d{\color{#3D99F6}b}+{\color{#20A900}c}+{\color{#69047E}d}
  • a+0+0.{\color{#D61F06}a} + {\color{#333333}0}+{\color{#333333}0}.

Are b,{\color{#3D99F6}b}, c,{\color{#20A900}c}, and d{\color{#69047E}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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