adidas and Puma did indeed split as they had been conflicting
personalities for quite awhile.
If you do want to know, adidas has been the major player in the
world of sports. Adolf Dassler, the founder of adidas, was the man
credited with helping Germany win their first world cup in The
Miracle of Bern. he is also the man responsible for the appointment
of the current Fifa president, Sepp Blatter.
adidas became the powerhouse they are today because of one man,
Adlof Dassler's son, Horst Dassler. Under Horst, adidas was split
internally for awhile, with the German side being under the control
of his parents and adidas international being aggresively expanded
by him. At one time, even Le Coq Sportif was under adidas.
As for Nike, they did enter the market because of Asics/ Tiger
Onitsuka. They started the company under the name Blue Ribbon
Sports in Oregon. Blue Ribbon Sports simply imported Asics/ Tiger
Onitsuka and were the sole distributors in America. The founder of
Nike then "stole" the technology and created some shoes of his own
and registered the name of the company as Nike.
It's interesting to note that while there are many rumors regarding
the vast sums of money paid to the designers of the iconic Nike
swish tick, the reality is that a design student was paid US$25 to
design it.
Also, in the 80s to 90s, adidas, though valuable as a brand, was
slowly fading due to weak leadership after the deaths of both Adolf
and Horst Dassler. It was the two men who helped market Nike to
become the powerhouse they are today that also helped to revive the
fortunes of adidas.
They are the men responsible for the two different logos of adidas,
namely the three stripes of increasing heights, as seen on their
football boots for example, and the trefoil, or the "old school"
adidas logo, which has now been used to brand their "traditional"
designs.