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232 8. Miscellaneous Problems

12. If x, y, z are rational numbers for which x3 + 3y3 + 9z3 - 9xyt. = 0,
    prove that x = y = z = 0.


13. Solve each of the following equations for unequal integers x, y:

(a) (x + 1)2 -x2 - (x - 1)2 = (y + 1)2 - y2 - (y - 1)2

(b) (x + 1)3 - z3 - (x - 1)s = (y + 1)s - ys - (y - 1)s

(c) (x + 1)4 - x4 - (x - 1)4 = (y + 1)4 - y4 - (y - 1)4.

14. Let the polynomial f(t) = t” + a,-it’+l +.a. + art + 1 have nonneg-
    ative coefficients and n real zeros. Prove that f(2) 1 3”.


15. Solve the system of equations:

x2 + 2yz = x

y” + 2x2 = %

22+2xy=y.

16. P, Q, R are three polynomials over R of degree 3 for which P(z) 5
    Q(x) 5 R( z ) f or a 11 real x. For some real U, equality holds. Prove that
    there exists a constant k with 0 5 k < 1 for which Q = ICP + (1 - k)R.
    Does this property still hold if P, Q, R are of degree 4?


17. Find an explicit polynomial P(a, b) such that there is a straight line
    intersecting the graph of y = x4 + ax3 + bx2 + cx + d in exactly four
    points if and only if P(Q, b) > 0.


18. Find the value of the real number p for which the equation

x3+px2+3x-lO=O

has three real roots a, b, c for which c - b = b - a > 0.

19. p(x) and q(x) are polynomials which satisfy the identity p(q(x)) =
    q(p(x)) for all real x. If the equation p(x) = q(x) has no real solution,
    show that the equation p(p(x)) = q(q(x)) also has no real solution.


20. Let P(x, y) be a polynomial in x and y of degree at most 2. Suppose
    that A, B, C, A’, B’, C’ are six distinct points in the xy-plane such
    that A’ lies on BC, B’ lies on AC and C’ lies on AB. Prove that if
    P vanishes at these six points, then P is identically equal to zero.


21. Show that
    c(a - b)3 + a(b - c)” + b(c - Q)~ = a + b + c
    c2(b-u)+a2(c-b)+b2(a-c)


22. Determine a polynomial with integer coefficients one of whose zeros
    is (3/5)‘i7 + (5/3)ii7.