Problem 4, 1999

Circles \(k_1, k_2, \ldots, k_{1999}\) lie in the plane and touch one another externally in a closed chain: \(k_1\) touches \(k_2\) at \(A_1\), \(k_2\) touches \(k_3\) at \(A_2\), and so on, and finally \(k_{1999}\) touches \(k_1\) at \(A_{1999}\).

Pick any point \(M_1\) on \(k_1\). Let \(M_2\) be the second point in which the line \(A_1M_1\) meets \(k_2\), let \(M_3\) be the second point in which the line \(A_2M_2\) meets \(k_3\), and continue in this way; the last step takes \(M_{2000}\) to be the second point in which the line \(A_{1999}M_{1999}\) meets \(k_1\). Prove that \(M_1\) and \(M_{2000}\) are the endpoints of a diameter of \(k_1\).