Given a natural number $k$ and a real number $a (a>0)$, find the maximal value of $a^{k_1}+a^{k_2}+\cdots +a^{k_r}$, where $k_1+k_2+\cdots +k_r=k$ ($k_i\in \mathbb{N} ,1\le r \le k$).
Source: China Mathematical Olympiad 1993 problem2
Tags: inequalities, inequalities unsolved
Given a natural number $k$ and a real number $a (a>0)$, find the maximal value of $a^{k_1}+a^{k_2}+\cdots +a^{k_r}$, where $k_1+k_2+\cdots +k_r=k$ ($k_i\in \mathbb{N} ,1\le r \le k$).