Problem of the week

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This topic contains 3 replies, has 1 voice, and was last updated by  Dr. Justin Albert 2 months, 4 weeks ago.

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  • #1794 Reply

    Squares in circle

    Here’s a problem a colleague had posted on the wall for students to work on. I thought it was fun, so I hoped you would to. Let me know your answers.

    If both objects are squares with the smaller square having endpoints on the circle equidistant from the angle bisector of angle BAC, what is the ratio of the larger square area to the smaller square area?

    #1960 Reply

    #2 A. Suppose that you are given a line of length 1. Using straight-edge and compass only do the following.
    a. Construct a line of length 5.
    b. Construct a line of length \(\sqrt{5}\).
    c. Construct a line of length \(\frac{1+\sqrt{5}}{2}\).

    Extra Credit

    B. Using part A (or by other techniques) construct a regular pentagon.

    C. Show that there is no distance that divides the length 1 and the length \(\frac{1+\sqrt{5}}{2}\).

    Note that you can use Geogebra in order to have a computer form of a straight-edge and compass. You can also learn more about this at Symmetry of regular polygons as well.

    #1962 Reply

    #3 Show that for any integer, \(n\), and any prime \(p\), \(n^{p}\equiv n\) mod \(p\). In other words, \((n^{p}-n)\) is divisible by \(p\).

    See Modular Arithmetic: 2+2=1! for more information on modular arithmetic.

    #2148 Reply

    #4 Suppose that in the following picture we have that \(AD=2DB\), \(BE=DB+2\), \(EC=DB+1\), \(CF=DB\) and \(AC=4-DB\). Find the lengths of \(AB,BC\) and \(AC\).

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