Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 6.0 National olympiad Prove it Silk Road Mathematics Competition

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

Solution

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.

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