(a) The \(n\)th term of an arithmetic progression is given by \(a_n = a + (n-1)d\), where \(a\) is the first term and \(d\) is the common difference.

Here, \(a = 84\) and \(d = -3\). We need \(a_n < 0\).

\(84 + (n-1)(-3) < 0\)

\(84 - 3(n-1) < 0\)

\(84 - 3n + 3 < 0\)

\(87 < 3n\)

\(n > 29\)

The smallest integer \(n\) is 30.

(b) The sum of the first \(m\) terms of an arithmetic progression is given by \(S_m = \frac{m}{2} [2a + (m-1)d]\).

We are given \(S_{2k} = S_k\).

\(\frac{2k}{2} [2 \times 84 + (2k-1)(-3)] = \frac{k}{2} [2 \times 84 + (k-1)(-3)]\)

\(k [168 + (2k-1)(-3)] = \frac{k}{2} [168 + (k-1)(-3)]\)

\(k [168 - 6k + 3] = \frac{k}{2} [168 - 3k + 3]\)

\(k [171 - 6k] = \frac{k}{2} [171 - 3k]\)

\(2k [171 - 6k] = k [171 - 3k]\)

\(342k - 12k^2 = 171k - 3k^2\)

\(342k - 171k = 12k^2 - 3k^2\)

\(171k = 9k^2\)

\(19k = k^2\)

\(k = 19\)