$(10$ points$)$ Prove or disprove that the product of a nonzero rational number and an

irrational number is irrational

$(10$ points$)$ Prove using contrapositive: If $4k^2+3k+2$ is odd then $k$ is odd.

$(10$ points$)$ Prove that for all ninN,

$1^2+4^2+7^2+{10}^2+$cdots$+(3n-2{)}^2=\frac{n(6n^2-3n-1)}{2}$

$(10$ points$)$ Show $n^3+2n$ is divisible by $3,$AAninN.

$(0$ points$)$ Show that it is possible to arrange the numbers $1,2,$dots,$n$ in a row so that

the average of any two of these numbers never appears between them.