2nd Annual
Virginia Tech Regional Mathematics Contest
From 9:30a.m. to 12:00 noon, November 8, 1980

Fill out the individual registration form

  1. Let * denote a binary operation on a set S with the property that

    (w*x)*(y*z) = w*z    for allw, x, y, z $\displaystyle \in$ S.

    Show
    (a)
    If a*b = c, then c*c = c.

    (b)
    If a*b = c, then a*x = c*x for all x $ \in$ S.

  2. The sum of the first n terms of the sequence

    1,    (1 + 2),    (1 + 2 + 22),...,(1 + 2 + ... + 2k - 1),...

    is of the form 2n + R + Sn2 + Tn + U for all n > 0. Find R, S, T and U.

  3. Let an = (1·3·5·...·(2n - 1))/(2·4·6·...·2n).
    (a)
    Prove that limn - > $\scriptstyle \infty$an exists.

    (b)
    Show that an = ((1 - (1/2)2)(1 - (1/4)2)...(1 - (1/2n)2))/((2n + 1)an).

    (c)
    Find limn - > $\scriptstyle \infty$an and justify your answer.

  4. Let P(x) be any polynomial of degree at most 3. It can be shown that there are numbers x1 and x2 such that $ \int_{-1}^{1}$P(x) dx = P(x1) + P(x2), where x1 and x2 are independent of the polynomial P.
    (a)
    Show that x1 = - x2.

    (b)
    Find x1 and x2.

  5. For x > 0, show that ex < (1 + x)1 + x.

  6. Given the linear fractional transformation of x into f1(x) = (2x - 1)/(x + 1), define fn + 1(x) = f1(fn(x)) for n = 1, 2, 3,.... It can be shown that f35 = f5. Determine A, B, C, and D so that f28(x) = (Ax + B)/(Cx + D).

  7. Let S be the set of all ordered pairs of integers (m, n) satisfying m > 0 and n < 0. Let < be a partial ordering on S defined by the statement: (m, n) < (m', n') if and only if m$ \le$m' and n$ \le$n'. An example is (5, - 10) < (8, - 2). Now let O be a completely ordered subset of S, i.e. if (a, b) $ \in$ O and (c, d ) $ \in$ O, then (a, b) < (c, d ) or (c, d ) < (a, b). Also let O denote the collection of all such completely ordered sets.
    (a)
    Determine whether an arbitrary O $ \in$ O is finite.

    (b)
    Determine whether the cardinality || O|| of O is bounded for O $ \in$ O.

    (c)
    Determine whether || O|| can be countably infinite for any O $ \in$ O.

  8. Let z = x + iy be a complex number with x And y rational and with | z| = 1.
    (a)
    Find two such complex numbers.

    (b)
    Show that | z2n - 1| = 2| sinnq|, where z = eiq.

    (c)
    Show that | z2n - 1| is rational for every n.





Peter Linnell
2001-08-17