TITLE: Choosing a menu for a dinner party
QUESTION [5 upvotes]: I heard of this puzzle that I want to get some hints or a full solution. 
There are $1024$ dishes to be chosen from for a party. There are $6875$ participants in total. The objective is to find $10$ dishes in the menu such that for any dish $d$ that is not on the list, there are more than half people who would prefer one of the dishes in the menu over $d$. 
Is it possible to generate such a menu?

REPLY [2 votes]: Have a tournament where each pair of dinners "fight," and the winner is the is the dinner which the majority of contestants prefer. There are $1024\cdot 1023/2$ fights, so there are that many wins. Since there are $1024$ dinners, each dinner wins an average of $1023/2=511.5$ fights. Some dinner must win at least that average many fights, so some dinner $d$ has won $512$ fights. Include that dinner in the menu; the $512$ dinners that $d$ will be off the menu, which is OK since they are preferred less than $d$. 
We are now left with a smaller power of two. Repeat this process with these $512$ dinners, then $256$, then $128$, etc, selecting one dinner at each level.