Let $m,nin{R}_{+},$ and $[*]$:$RZ$ be the floor function.

Question $1(2$ pts$).$ Prove that $[m]+[n][m+n][m]+[n]+1.$

Hint: Write $m=[m]+\{m\},n=[n]+\{n\},$ where $[m],[n]inZ$ and $\{m\},\{n\}in[0,1).$

Question $2(2$ pts$).$ Based on the previous sub$-$question, prove that if $x,$yinN s$.$t$.x$ is $k-$bits

and $y$ is $l-$bits, then their products $xy$ is at most $(k+l)-$bits.

Question $3(2$ pts$).$ Based on the previous sub$-$question, prove that if ninN is $k-$bits, then

AAdinN,$n^d$ is at most $dk-$bits.

Question $4(4$ pts$).$ Based on the previous sub$-$question, prove that for $m,$ninN,

$T(C_n^m)=O(m^2(lo{g}_{2}n{)}^2)$