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Miscellaneous Problems 235

35. Show that the set of real numbers x which satisfy the inequality
    70
    k
    c- x-k >2 - 4
    k=l

is a union of disjoint intervals, the sum of whose lengths is 1988.

36. Let a, b, c, d, e, f be complex numbers for which Ca = Ca3 = 0.
    Prove that

(a + c)(a + d)(a + e)(a + f) = (b + c)(b + d)(b + e)(b + fh

37. Eliminate u, v, w from the equations

a=cosu+cosv+cosw

b = sin u + sin v + sin w

c=cos2u+cos2v+cos2w

d = sin 2u + sin 2v + sin 2w.

38. Eliminate 0 between
    x=cotB+tan0
    y=secB-cose.


39. Show that x6 - x5 + x4 - x3 + x2 - x + 314 has no real zeros.


40. Prove that the local maximum and local minimum values of the real
    polynomial x3 + 3px2 + 3qx + r are given by

2p3 - 3pq + P + 2(p2 - q)3f2

and

2p3 - 3pq + r - 2(p2 - q)3’2.

41. (a) Suppose that

a2 + b2 - c2 - d2 a2 - b2 - c2 + d2

=

a-b+c-d a+b+c+d ’

Show that

ab - cd bc-ad

a-b+c-d=a+b+c+d’

(b) Find a set of integers a, b, c, d which satisfy the equations in

(4.

42. Prove that a real polynomial p( x ) w h ic h assumes rational values for
    rational x and irrational values for irrational z must be linear.