#### Japan

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Let $p$ be an odd prime number. For positive integer $k$ satisfying $1\le k\le p-1$, the number of divisors of $k p+1$ between $k$ and $p$ exclusive is $a_k$. Find the value of $a_1+a_2+\ldots + a_{p-1}$.

Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that $$f(yf(x)-x)=f(x)f(y)+2x$$ for all $x,\ y\in{\mathbb{R}}$.