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# Problems of the Week

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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?

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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