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154 4. Equations

29. Given that a 2 1 and b are real numbers, prove that the system

y=x3+ax+b

z=y3+ay+b

x=z3+az+b

has exactly one real solution.

30. Solve
    x(x - y - z) = a

Y(Y - z-x)=b

I(% - x - y) = c.

31. If x + x-l = a, y + y-l = b and a2 + b2 + z2 = 4 + abz, determine z
    in terms of x and y.


32. If

and

ux vy W%

show that (x/u + y/v + .z/w)~ = a2/u2 + b2/v2 + c2/w2.

33. Let a, b, c be positive numbers such that fi + 4 + fi = a/2.
    Prove that the system

Jy--a+JTT=l

Jzi;+&-S=1

JKT+Jy-c=1

has exactly one real solution.

34. Solve for 2, y, z:
    yz = a(y + z)
    xz = b(x + z)

xy = c(x + y).