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Miscellaneous Problems 237

53. A triangle has sides of length 29, 29, 40. Find all other triangles with
    integer sides with the same perimeter and area.


54. Show that

Nx - Y>(X - 2) + (Y - %)(Y - x) + t% - XX% - Y)l

can be expressed as the sum of three squares.

55. If x, y, % > 0, show that
    -^1 -^1 -^1 ~^3
    x(1 + y) + y(l+ %) + z(l+ x) > 1+xyt’


56. Reduce to lowest terms
    (ab - x2)2 + (ax + bx - 2x2)(ax + bx - 2ab)
    (ab + x2)2 - x2(a + b)2

Explorations

E.62. For n 2 2, let qn(z) be the polynomial %-‘[(l + z)” - 1 - %“I. For
which values of n do all its zeros satisfy ]%I = l?

E.63. Two Trigonometric Products. For small positive integer values
of m, use a pocket calculator to find the approximate values of the products

cos(n/(2m + 1)) cos(27r/(2m + 1)) cos(37r/(2m + 1))... cos(m?r/(2m + 1))

and

tan(s/(2m + 1)) tan(27r/(2m + 1)) tan(37r/(2m + 1))... tan(m7r/(2m + 1)).

Make a conjecture as to the exact values of these expressions. Can you
establish this conjecture, at least for some values of m?
One way to approach the situation is to look at the polynomial

f(t) = fi (t - set” &).

k=l

The two products can be found by taking the square roots of the values

If(O)] and If(l)]. With the help of a calculator, make conjectures regard-

ing the coefficient of f(t) f or various values of m, and try to check your

conjecture for as many cases as possible.

E.64. Polynomials All of Whose Derivatives Have Integer Zeros.

The cubic polynomial t3 - 36t2 + 285t - 250 and its first and second deriva-

tives all have integer zeros. Find other cubic polynomials with the same