Seven faculty members at a management institute frequent a lounge for strong coffee and stimulating conversation. On being asked about their visit to the lounge last Friday we got the following responses.
JC:I came in first, and the next two persons to enter were SS and SM. When I left the lounge, JP and VR were present in the lounge. DG left with me.
JP:When I entered the lounge with VR, JC was sitting there. There was someone else, but I cannot remember who it was.
SM: I went to the lounge for a short while, and met JC, SS and DG in the lounge that day.
SS: I left immediately after SM left.
DG: I met JC, SS, SM, JP and VR during my first visit to the lounge, I went back to my office with JC. When I went to the lounge the second time, JP and VR were there.
PK: I had some urgent work, so I did not sit in the lounge that day, but just collected my coffee and left. JP and DG were the only people in the lounge while I was there.
VR: No comments.
For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is: