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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?

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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