Silk Road Mathematics Competition

Question

Let a_1, a_2, , a_2003 be a sequence of real numbers. A term a_k, 1 k 2003, is said to be a *leading term*, if at least one of the expressions a_k, a_k+a_k+1, , a_k+a_k+1++a_2003 is positive. Prove that the sum of all leading terms is positive provided that the sequence has at least one leading term.

Accepted Answer

We solve this problem for any sequence having

n

terms applying induction with respect to

n

.

The case

n = 1

is clear.

Suppose that the statement is true for all sequences of length less than

n

.

Now consider a sequence

a_{1}, a_{2}, \dots, a_{n}

.

Case 1. $a_{1}$ is not a leading term.

Then the set of all leading terms of the sequence

a_{1}, a_{2}, \dots, a_{n}

coincides with the set of all leading terms of the sequence

a_{2}, a_{3}, \dots, a_{n}

. And by inductive hypothesis we are done.

Case 2. $a_{1}$ is a leading term.

Consider the smallest nonnegative integer

m

, with positive

a_{1} + a_{2} + \dots + a_{m}

. Then the terms

a_{2}, a_{3}, \dots, a_{m}

are also leading terms and their sum is positive. The sum of all remaining leading terms also is nonnegative by induction hypothesis.