and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
-
in which each term is either 0 or 1 is called a binary sequence of length 
be the number of binary sequences of length 
be the number of binary sequences of length 
for all positive integers 
points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of 
Y by
Let 
be a natural number. At a mathematical olympiad training camp the same 
courses are organised every day. Each student takes exactly one of the courses each day. At the end of the camp, every student has takes each course exactly once, and any two students took the same course on at least one day, but took different courses on at least one other day. What is, in terms of , the largest possible number of students at the camp?
be a natural number. At a mathematical olympiad training camp the same 
courses are organised every day. Each student takes exactly one of the courses each day. At the end of the camp, every student has takes each course exactly once, and any two students took the same course on at least one day, but took different courses on at least one other day. What is, in terms of , the largest possible number of students at the camp?
for the courses and everyone takes the final course together, and for a bound note that if we cyclically shift a permutation of courses it doesn't have an intersection with itself, so out of every cycle we have at most 1 permutation and there are 
.
on the first day. Then, for days 
,
,
.
students, and we're done.
