Reach 180

First I ll prove how the first person to 90 will win , and then use the same logic repeatedly.

Once I reach 90 , my opponent must at least speak 91 and thus allowing me to speak 180 .He can NOT say 180 before this , as the largest before 90 could have been 89 , and thus he can at most say 178 , which means irrespective of what he says I will be able to say 180 next .

Following a similar logic we can say that first to 45 wins .

Now we must consider 22 or 23 , well we can check it , if I say 22 then my opponent can say 23 , which will mean the maximum number that I can say is 44 , which means he ll win . So first to 23 wins NOT 22 .

Following the same logic , first to 12 wins , then first to 6 and then first to 3 , and we already know that this is Sam .

The numbers which a person can force are like : 2,3,4,(5...6),8,(9...13),16,(17...28)

Let the 2 consecutive numbers be $x$ & $y$ . $x<y$

We can easily say that $2x<x+y$ or $2x$ , $x$ & $y$ will form a triangle irrespective of $y$ (provided y comes after $x$ in the sequence)

So if a person can guess $x$ he/she can guess $2x$ . Also by similar logic , he/she can also guess $2x-1$ (Provided other player cannot guess it first, can be easily shown invalid in this case.) This suggests that if he/she needs to guess $2x-1$ , he/she needs to guess $x$ which means halved and rounded up!

All that remains is to go on halving 90=>45=>23=>12=>6=>3 so whoever says 3 will win i.e Sam