SELECTION and TRAINING SESSION

Answer: yes, such numbers exist. Consider the triplets $(a,b,c)=(13,4,3)$ and $(d,e,m)=(12,7,1)$. These numbers satisfy the system $$\{\begin{array}{l}a+b+c=20=d+e+m,\\ ab+bc+ac=103=de+em+dm,\\ abc-dem=3^2\cdot 2^3.\end{array}$$ Clearly, for $k=2^{673}\cdot 3^{673}$ the numbers $ak$, $bk$, $ck$, $dk$, $ek$ and $mk$ satisfy the problem conditions.