# Problems of the Week

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# 2018-06-18 Intermediate

An infinite plane can be tiled using identical triangles, quadrilaterals, or hexagons with no overlaps and no gaps.

Is it possible to tile an infinite plane using identical pentagons?

A solid disk, a solid sphere, and a hoop of identical mass $$m$$ and radius $$R$$ are mounted at rest on frictionless axles. Each object has a string wrapped around its circumference, and the strings are each pulled with an identical force $$F$$ for the same time interval; the strings do not slip.

Which object will have the greatest rotational kinetic energy at the end of the time interval?

Note: The mass of the spokes on the hoop is negligible.

Lines are drawn through point $$P$$ parallel to the sides of triangle $$ABC.$$ The resulting 3 triangles have areas 4, 9, and 49.

What is the area of triangle $$ABC?$$

The areas of these 3 smaller triangles are 4, 9, and 49. What's the area of the big, outer triangle?

In this rectangular window frame, a spider begins to weave its web. The basic framework consists of three threads, which start from points $$A, B,$$ and $$C$$ and are glued together at point $$D.$$

What is the length $$l_A$$ of the first thread in centimeters?

Assumptions: Without tension, all threads have the same length, but they are stretched many times their original length. Thus, the tension force $$F_i \approx - k l_i$$ acting along a thread is approximately proportional to its total length $$l_i$$. While the lengths $$l_i$$ of the threads can be different, the spring constant $$k$$ is the same for all threads. Apart from the tension, no further forces act on the spider threads.

$$a,b,$$ and $$x$$ are real numbers satisfying the following system of equations: $\begin{eqnarray} a+b &=& 3x-2 \\ ab &=& 4x^2-3x-4. \end{eqnarray}$ What is the minimum value of $$a^2+b^2$$ (to three decimal places)?

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