BdMO national Junior 2010/6

[samiul_samin]

Nonte has been given all the numbers from $1$ to $40$ & Fonte has been given all the odd numbers from $1$ to $100$.They have to take any two given numbers so that the summation of those $2$ numbers is divisible by $3$.He,who will make the numbers divisible by $3$ in most ways,will be announced winner.Who will be the winner and why?

[samiul_samin]

Answer:$\fbox {Fonte}$

Solution:
\[\lfloor{\dfrac{50}{3}}\rfloor+1=14\]

\[\lfloor{\dfrac{40}{3}}\rfloor+1=17\]

So fonte can make $\dbinom {17}{2}=136$ choices.

So nonte can make $\dbinom {14}{2}=91$ choices.

As,\[136>91\]

Fonte will be the winner.