[/quote]
,
,
.
and 
Prove that
with circumcenter 
and orthocenter 
,
be an arbitrary point on the circumcircle of triangle 
and that line 
is not parallel to line 
.
be the point on the circumcircle of triangle 
is perpendicular to 
be the point on line 
such that 
.
and 
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
be nonnegative real numbers such that 
.
are natural numbers satisfy that: 
and 
are prime numbers. Prove that 
is a perfect squared number
Prove that
denote the sum of the sum of the factors of all positive 
,
is the sum of the factors of 
added to the sum of the factors of 
and so on until the sum of the factors of 
,
.
denote 
.
or find a counterexample
such that
be nonnegative real numbers such that 
Prove that
Let 
Prove that
Y by Adventure10, Mango247
Two players A and B play the following game. Initially, there are 
equal positive integers 
written on a blackboard. A begins and the players move alternately. The player to move chooses one of the non-zero numbers on the board. If this number k is the smallest among all positive integers on the board, the player replaces it with 
; if not, the player replaces it with the smallest positive number on the board. The player who first turns all the numbers into zeroes, wins. Who wins if both players use their best strategies?
equal positive integers 
written on a blackboard. A begins and the players move alternately. The player to move chooses one of the non-zero numbers on the board. If this number k is the smallest among all positive integers on the board, the player replaces it with 
; if not, the player replaces it with the smallest positive number on the board. The player who first turns all the numbers into zeroes, wins. Who wins if both players use their best strategies?
.
,
,
.
