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Re: So you think you know everything?

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  • jwwarrenva
    ... There are 293 ways to make change for a dollar, but only if you include a silver dollar among your coins. Here is a generatingfunctionological approach.
    Message 1 of 7 , Jul 1, 2007
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      --- In mathforfun@yahoogroups.com, "cino hilliard" <hillcino368@...>
      wrote:
      >
      > Take a look at
      >
      > http://www.usaone.net/jokenet/jokes.asp?command=list&r=108
      >
      > Maybe someone here would like to prove number 3.
      >
      > Have fun,
      > Cino
      >

      There are 293 ways to make change for a dollar, but only if you
      include a silver dollar among your coins. Here is a
      generatingfunctionological approach.

      Let
      P100 = 1 + x^100
      P50 = 1 + x^50 + x^100
      P25 = 1 + x^25 + x^50 + x^100
      P10 = 1 + x^10 + x^20 + … + x^100
      P5 = 1 + x^5 +x^10 + … + x^100
      P1 = 1 + x + x^2 + … + x^100
      Then the number you want (293) is the coefficient of x^100 in P1 * P5
      * P10 * P25 * P50 * P100.
      In the book "Concrete Mathematics", it is shown how this idea can be
      extended to produce a formula for the number of ways to make change
      for N cents.
    • Dora
      Studies show goldfish have memories of more than 3 seconds. The speaker of the house can speak. He can t take sides in a debate, he effectively chairs the
      Message 2 of 7 , Jul 1, 2007
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        Studies show goldfish have memories of more than 3 seconds.

        The speaker of the house can speak. He can't take sides in a debate, he
        effectively chairs the debate. But he can and does speak.

        I've doubts about lots of the rest of these too.

        Dora



        [Non-text portions of this message have been removed]
      • Daniel
        For A + B + C = 360 degrees Prove that [sin(A) + sin(B) + sin(C)] is maximum when A = B = C I kind of see a pattern on excel in favor the proof but just
        Message 3 of 7 , Jul 1, 2007
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          For A + B + C = 360 degrees

          Prove that [sin(A) + sin(B) + sin(C)] is maximum when A = B = C



          I kind of see a pattern on excel in favor the proof but just wondering if
          there is a conclusive derivation.



          Daniel




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        • video_ranger
          ... wondering if ... Except for a factor (1/2) this is the area of the triangle formed by three points on a unit circle where A,B,C are the arc angles between
          Message 4 of 7 , Jul 2, 2007
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            --- In mathforfun@yahoogroups.com, "Daniel" <dbeyene@...> wrote:
            >
            > For A + B + C = 360 degrees
            >
            > Prove that [sin(A) + sin(B) + sin(C)] is maximum when A = B = C
            >
            >
            >
            > I kind of see a pattern on excel in favor the proof but just
            wondering if
            > there is a conclusive derivation.
            >
            >
            >
            > Daniel

            Except for a factor (1/2) this is the area of the triangle formed by
            three points on a unit circle where A,B,C are the arc angles between
            pairs of the three radii from the center to the three points (the
            area of a triangle formed by two radii of unit length with angle x
            between them is (1/2)sin(x)).

            --> The problem is equivalent to proving that if three points on a
            circle are chosen to maximize the area of their triangle, it must be
            equilateral (A = B = C = 120).

            Suppose P1,P2,P3 are three points on a circle forming a triangle of
            maximal area and that two sides, say P1P3 and P1P2 are not equal.
            Then the point T on the circle at the top of the arc of the chord
            P2P3 is farther from the chord P2P3 than P1, so the triangle TP2P3
            has the same base (P2P3) as P1P2P3 but a higher altitude --> its area
            is bigger, contradicting the assumption.
          • clooneman
            And look at No 8. They forgot stretched and scratched .
            Message 5 of 7 , Jul 20, 2007
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              And look at No 8. They forgot "stretched" and "scratched".

              --- In mathforfun@yahoogroups.com, "cino hilliard" <hillcino368@...>
              wrote:
              >
              > Take a look at
              >
              > http://www.usaone.net/jokenet/jokes.asp?command=list&r=108
              >
              > Maybe someone here would like to prove number 3.
              >
              > Have fun,
              > Cino
              >
            • Peter Otzen
              ... And No. 28 is rubbish. Such a statement only provides the incentive to sneeze WITHOUT closing your eyes. Excellent practice when driving a car in city
              Message 6 of 7 , Jul 21, 2007
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                --- In mathforfun@yahoogroups.com, "clooneman" <clooneman@...> wrote:
                >
                > And look at No 8. They forgot "stretched" and "scratched".
                >

                And No. 28 is rubbish. Such a statement only provides the incentive
                to sneeze WITHOUT closing your eyes. Excellent practice when driving
                a car in city traffic. Not all that hard either!!

                Peter

                > --- In mathforfun@yahoogroups.com, "cino hilliard" <hillcino368@>
                > wrote:
                > >
                > > Take a look at
                > >
                > > http://www.usaone.net/jokenet/jokes.asp?command=list&r=108
                > >
                > > Maybe someone here would like to prove number 3.
                > >
                > > Have fun,
                > > Cino
                > >
                >
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