#### Inequality Putnam

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Find all positive integers $n$ and $k_i$ $(1\le i \le n)$ such that $$k_1 + k_2 + \cdots + k_n = 5n-4$$ and $$\frac{1}{k_1} + \frac{1}{k_2} + \cdots + \frac{1}{k_n}=1$$

$$|\sin x + \cos x + \tan x + \cot x + \sec x + \csc x|$$ where $x$ is a real number.

Let $m$ and $n$ be positive integers. Show that $$\frac{(m+n)!}{(m+n)^{m+n}}<\frac{m!}{m^m}\frac{n!}{n^n}$$