Question 10
For any positive real number $x$, we define $[x]$ to be the greatest integer that does not exceed $x$ and $\{x\}$ to be the fractional part of $x$ such that $\{x\}=x-[x]$. Let $a$, $b$ and $c$ be positive real numbers that satisfy the following system of equations:
$$\begin{cases}a+2[b]+2\{c\}=4.6,\\\left[a\right]+\{b\}-c=1,\\2\{a\}+2b+[c]=4.5.\end{cases}$$
What is the largest possible value of $a+b+c$?
$$\begin{cases}a+2[b]+2\{c\}=4.6,\\\left[a\right]+\{b\}-c=1,\\2\{a\}+2b+[c]=4.5.\end{cases}$$
What is the largest possible value of $a+b+c$?