... Let A = (a1, a2, . . . , a 2001 ) be a sequence of positive integers. Let mbe the number of 3-element subsequences (ai, aj, ak) with 1 ≤ i < j < k ≤ 2001, suchthat aj= ai+ ... ai, . . . , a 2001 ).(2) If ai+1= ai+ 1 + d, where d > 0, increase a1, . . . , aiby d toobtain the new sequence (a1+d, a2+d, . . . , ai+d, ai+1, . . . , a 2001 ).It is clear ... partition of 2001 into positiveintegers t1, . . . , ts.The maximum value of m occurs when s = 3 or s = 4. If s > 4 then we mayincrease the value given by (∗) by using a partition of 2001 into...