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 $$\in$$ 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 + ... + 2^{k - 1}),...
   is of the form 2^{n + R} + Sn² + Tn + U for all n > 0. Find R, S, T and U.

3. Let a_n = (1·3·5·...·(2n - 1))/(2·4·6·...·2n).

   (a)

   Prove that lim_{n - > $\infty$}a_n exists.

   (b)

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

   (c)

   Find lim_{n - > $\infty$}a_n 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 e^x < (1 + x)^{1 + x}.

6. Given the linear fractional transformation of x into f₁(x) = (2x - 1)/(x + 1), define f_{n + 1}(x) = f₁(f_n(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 = e^iq.

   (c)

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