A numerical inequality. If $a_1\ge a_2\ge \cdots \ge a_n\ge 0$, $b_1\ge b_2\ge \cdots \ge b_n\ge 0$ with $\sum_{j=1}^kb_j=1$ for some $1\le k\le n$. Is it true that $2\sum_{j=1}^na_jb_j\le  a_1+\frac{1}{k}\sum_{j=1}^na_j$?
The above question is denied.
Give a simple proof to a weaker version? 
Under the same condtion, then $\sum_{j=1}^na_jb_j\le \max\{a_1,\frac{1}{k}\sum_{j=1}^na_j\}$.

No, it isn't true. Let $a_j=b_j=1$ for $1\leq j\leq n$. Then $\sum_{j=1}^1b_j=1$, hence $k=1$, thus we've satisfied the hypotheses. However, $$2\sum_{j=1}^na_jb_j=2n\geq1+n$$ with equality if and only if $n=1$.
EDIT: For the second case, let $a_j=1/10$ for all $j$, $b_1=1$ and $b_j=10$ for $j\geq 2$. Then we end up with $$n-0.9\geq n/10$$ with equality if and only if $n=1$ again. 
