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Problems of the Week

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$$ABCD$$ is a quadrilateral with $$AD=BC,$$ $$AB=40,$$ $$CD=20,$$ and $${\color{blue}m\angle A} + {\color{orange}m\angle B} = 90^\circ.$$

What is the area of quadrilateral $$ABCD?$$

Hint: Consider how copies of $$ABCD$$ can be constructed into another shape.

Let $$x,$$ and $$y,$$ and $$z$$ be real numbers satisfying $$\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{x+y}=1.$$

Find the maximum value of $$\dfrac{x^2}{y+z}+\dfrac{y^2}{z+x}+\dfrac{z^2}{x+y}.$$

Tom is jumping around a $$4\times4$$ grid of circles with the following conditions:

• He starts from any of the 16 circles.
• He always jumps in a straight line from one circle to any other circle.
• Each subsequent jump is farther than the last.
• He doesn't visit a circle already visited.

The maximum number of circles that he can step on is $$m,$$ and the number of ways of doing so is $$r.$$

Find the value of $$mr.$$


Note: The grid is fixed on the ground, so a method obtained by rotating another is considered different.

$$128=2^7,$$ but none of the other permutations of the digits of 128 form powers of 2: $182,\ 218,\ 281,\ 812,\ 821.$

Is there any power of 2, $$2^n,$$ such that at least one of its other permutations is also a power of 2?

A uniform disk sits on a smooth floor with a rocket strapped to its perimeter. One second after the rocket ignites, how far is the disk's center from where it started, in meters?


Details and Assumptions:

• The mass of the disk is $$M=\SI{1}{\kilo\gram},$$ its radius is $$r=\SI{0.5}{\meter},$$ and the rocket provides a constant thrust of $$T = \SI{10}{\newton}$$ after it ignites.
• Neglect the mass and size of the rocket.
• Submit your answer to to 3 decimal places.
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