Balancing Partial Sums

★ 7/10Abel Summation sequences inequality Casework

Problem

A sequence $\{a_n\}$ has $a_1 = 0$ , and $a_n - a_{n-1} \in \{1, 2\}$ for every integer $n \geqslant 2$ . Suppose a positive integer $k \in [1, 9]$ is such that

a_1 + a_2 + \cdots + a_k \leqslant a_{k+1} + a_{k+2} + \cdots + a_{10}

holds for every such sequence. Find the maximum value of $k$ .

Answer

Solution

Difficulty	7/10
Topics	Abel Summation, sequences, inequality, Casework

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