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236 8. Miscellaneous Problems

43. If a/(bc - u2) + b/(ca - b2) + c/(ab - c2) = 0, prove that
    a/(bc - a2)2 + b/( ca - b2)2 + c/(ab - c2)2 = 0.


44. Solve 1 + l/l + l/l + l/l + .e e + l/l + x = x. (The left side is a
    continued fraction with n slashes.)


45. Find all polynomials f(t) over R such that

f(t)@ + 1) = f(t2 + i? + 1).

46. Is there a set of real numbers u, v, w, x, y, % satisfying

u2 + v2 + w2 + 3(x2 + y2 + z”) = 6

47. Suppose that x + y + % = x-l + y-l + z-l = 0. Show that

x6 + y6 + 26

x3 + y3 + 23

= XY%.

48. Show that, if p is an odd prime and k is a positive integer, then

%P + 1 I(%@ - l)(%P--2 - %P-3 + * * * + % - l)k + (% + l)z(p-l)k-‘.

49. Observe that B3 - 73 = (22 + 32)2 and 1O53 - 1O43 = (g2 + 102)2.
    Show that, if the difference of two consecutive cubes is a square, then
    it is the square of the sum of two successive squares.


50. Prove that the only positive solution of

x+$+%3=3

y+%2+x3=3

z+x2+y3=3

is (x, y, %) = (1, l,l).

51. How many real solutions are there to the equation

6c2 - 77[2] + 147 = 0,

where [x] denotes the greatest integer not exceeding x?

52. Let k be a real parameter. Sketch the possible forms of the graphs of
    the equation
    y = x4 + 1x2 - 2k2(2k + 1)x,
    specifying for each form the values of k which give rise to it.