Answer: 22 minutes


In terms of the walking time
walk                             run                               cycle
          $\rightarrow2\times$ as fast               $\rightarrow\frac32\times$ as fast
          $\rightarrow\frac12$ the time                $\rightarrow\frac23$ the time 
   $w$                             $\tfrac12w$                               $\tfrac23\times\tfrac12w=\tfrac13w$

$\begin{align} w + \tfrac12w + \tfrac13w &= 3\times \tfrac13w+10\\
\Rightarrow \tfrac56w&=10\\
\Rightarrow w&=12\end{align}$
Total time: $12+\tfrac12 12 + \tfrac13 12 = 22$ minutes



In terms of the cycling time
Let the athlete take $x$ minutes to cycle one mile.

Therefore he takes $\frac{3}{2}x$ minutes to run one mile and $3x$ minutes to walk one mile.

So $3x+\frac{3}{2}x+x=3x+10$, i.e. $x=4$.

Therefore the cyclist takes $12$ minutes to walk the first mile, $6$ minutes to run the second mile and $4$ minutes to cycle the third mile. So the total time taken to walk, run and cycle the three miles is $22$ minutes.