Problema del lunes en martes

Ayer olvidé poner problema. Pero aquí están estos: 1) Let S be a set of n persona such that: (i) any person is acquainted with exactly k other persons in S; (ii) any two persons that are acquainted have exactly l common acquaintances in S; (iii) any tío persons that are not acquainted have exactly m common acquaintances in S. Prove that m(n-k) - k(k-l) + k - m = 0. 2) Let a_1 <= a_2 <= ... <= a_n = m be positive integres. Denote bu b_k the number of those a_i for which a_i >= k. Prove that a_1 + a_2 + ... + a_n = b_1 + b_2 + ... + b_m. 3) Let n be an odd integre greater than 1 and let c_1, c_2, ..., c_n be integers. For each permutation a = (a_1, a_2, ..., a_n) of {1, 2, ..., n}, define S(a) = sum of (c_i)(a_i) from i=1 to n. Prove that there exist not equal permutations a and b of {1, 2, ..., n} such that n! is a divisor of S(a) - S(b).