Since every player played with each other exactly once, there were $7+6+\dots+1=28$ matches in total. Since each tournament has $7$ matches, there were $28/7 = 4$ tournaments. a. Assume for the sake of contradiction that there was a player that only participated in one semi final. Then he would play at most $6$ matches, which is impossible since every player played $7$ matches. Hence every player must have played more than one semifinal. b. Assume for the sake of contradiction that there was one player that did not play in a final. Then this player must have played in $3$ semi-finals to be able to play $7$ matches. However if he played in $3$ semi finals then there must be one player who only played in one (since there are $16$ spots in the $8$ semi finals), which we know to be a contradiction from part a. Hence every player played in a final.