Finding best player,second best and third using least games possible. Note:assume if arm wrestler a beasts b and b beats c then a beats c.
Suppose you have an arm wrestling championship and 32 arm-wrestlers. What is the minimum matches you need to organizeto find the best second best and third best arm-wrestlers. The answer is supposed to be 39 but I can only manage to get it down to 42. This is how I did it: 
Make a round robin to find champ(31 uses)
the second place was beaten by champ, so find the best out of the 5 players champ beat (4 uses)
the champ was beaten by second or third.(second beat at most 4 other people)
find best of the people the champ defeated except the second(3). Find the best the second defeated(3). Put the best of those to find third place (1)
 A: I'm pretty sure the OP meant to say single-elimination, rather than round robin, tournament.  Let's say the best wrestler beats $A$, $B$, $C$, $D$, and $E$, in that order.  We know the second best wrestler must be one of these, so let them compete in a secondary tournament in the following order:  $A$ against $B$, the winner of that against $C$, the winner there against $D$, and the winner there against $E$.  These $4$ matches reveal the second best wrestler and, more important, we know that the third best is among the wrestlers that lost in direct competition to that one, either in the original tournament or in the secondary tournament.
There may be a slicker way of finishing things off, but here's a straightforward case-by-case approach:  $A$ beat no one in the original tournament and beats at most $4$ opponents in the secondary round; $B$ beat one wrestler in the original and at most $4$ in the secondary round; $C$ beat $2$ in the original and at most $3$ in the secondary; $D$ beat $3$ and at most $2$, and $E$ beat $4$ and at most $1$, so whichever wrestler is second best, the final competition, for third best, involves at most $5$ wrestlers, hence can be determined in $4$ matches.  
So all in all, identifying the best, second best, and third best wrestlers takes at most $31+4+4=39$ matches.  The real key is the order in which the secondary tournament takes play.
Added later:  Note, I haven't shown that $39$ is the minimum number, just that you can do it in at most $39$ (as opposed to $42$) matches.
