AoPS Academy
people be arranged in a circle, numbered clockwise from 
to 
. These people are eliminated one by one in order, until only one person remains. The elimination follows this rule: among the remaining people, start counting clockwise from the person with the smallest number, and eliminate the -th person in that count. Then, among the remaining people, start counting again from the person with the smallest number and eliminate the -th person. Repeat this process until only one person remains. Let 
denote the number of the last remaining person.
, people are eliminated in the following order: 
. Thus, 
. It is known that 
under certain conditions. Prove that the necessary and sufficient condition for this is that both 
and 
are prime numbers.
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer such that 
is called the period of 
. Determine the total number of points in 
of period 
.
)
and 
and 
consecutive even numbers, such that the sum of their squares divides the square of their product.
be positive integers such that 
and 
. Find the maximum possible value of 
.
, and this is done by finding the minimum value of 
, 
gives the maximum which yields:
i.e., 
, and so we solve to get 
.
of their two-point shots and 
of their three-point shots, which resulted in 54 points. They attempted