Let S be the set {a ϵ Z<sup>+</sup> : a ≤ 100}. If the equation [tan<sup>2</sup> x] - tan x - a = 0 has real roots (where [.] is the greatest integer function), then the number of elements is S is
\([\tan^2 x] = \tan x + a\). The left side is an integer and a is an integer, so \(\tan x = m\) must be an integer. Then \([\tan^2 x] = m^2\).
\(m^2 = m + a \Rightarrow a = m(m - 1)\)
With \(1 \le a \le 100\): \(a = 2, 6, 12, 20, 30, 42, 56, 72, 90\) (for m = 2 to 10).
That gives 9 values.
Answer: (C) 9
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