Unknown

234 8. Miscellaneous Problems

27. Determine necessary and sufficient conditions on a, b, c such that

az+by+cz=O

and

a~=+b&-&ccJ1-z2=0

should admit a real solution x, y, Z, with 1x1 5 1, IyI 5 1, 1.~1 5 1.

28. Let the polynomial xl’+xg+.. -+x + 1 be given, where the starred
    coefficients are to be filled in by two players playing alternately until
    no stars remain. The first player wins if all zeros of the polynomial are
    nonreal; the second player wins otherwise. Is there a winning strategy
    for the second player?


29. Let A, B, X, Y be variables subject to AX - BY = 1.

(a) Find explicit polynomials u and v in A, B, X, Y over Z such

that A4u - B4v = 1.

(b) Show that, for each pair m, n of positive integers, there are

polynomials u, v in A, B, X, Y for which A”‘u - B”v = 1.

30. Let p(x) be a polynomial over R of even degree n for which p(x) > 0
    for all x. Prove that p(x) + p’(x) +... + p(“)(x) > 0 for all x.


31. Three positive numbers 2, y, z lie between the least and greatest of
    three positive numbers a, b, c. If

x+y+z =a+b+c and xy.~ = abc,

show that, in some order x, y, z are equal to a, b, c.

32. Prove that every polynomial has a nonzero polynomial multiple whose
    exponents are all divisible by 1 000 000.


33. Let p and q be polynomials over C of positive degree. Suppose that
    Pk = {Z E C : p(z) = k} and Qk = {Z E C : q(r) = k}. Show that,
    if PO = Qc and PI = Qi, then p = q.


34. Suppose that U, v, w, x, y, z are real numbers with x, y, z all distinct
    for which the equations
    u3 + x3 = 113 + y3 = w3 + r3 = Q3

and

u(y - %) + V(% - x) + w(x - y) = 0

hold. Show that uvw + xyz = a3.

What is the situation if x = y?