[/quote]
,
,
is a perfect square? (Here 
denotes the integer part of the real number 
?
satisfying
![\[
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2^a + 3^b + 5^c = n!
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of injective functions from the set of real numbers to itself satisfying
and 
.
is injective if 
for any distinct real numbers 
and 
.
,
such that any graph with 
{
}
distinct positive integers is 
if for every 
the number 
is divisible by the product of all numbers in 
.
positive integers.
with 
and 
.
?
is a cyclic quadrilateral inscribed in a circle of radius one unit. If 
,
be the Fermat point of a 
.
,
,
are concurrent and the point of concurrence is 
,
,
?
.
and 
externally on the sides 
and 
respectively.
meets 
at 
,
meets 
at 
and 
denotes circumcenter of 
and 
meets again on 
.
Y by
Let 
and 
be positive integers. Kellem and Carmen play the following game: initially, the number 
is on the board. Starting with Kellem and alternating turns, they add powers of to the previous number on the board, such that the new value on the board does not exceed . The player who writes wins. Determine, for each pair 
, who has the winning strategy.
Note: A power of is a number of the form 
, where 
is a non-negative integer.
and 
be positive integers. Kellem and Carmen play the following game: initially, the number 
is on the board. Starting with Kellem and alternating turns, they add powers of to the previous number on the board, such that the new value on the board does not exceed . The player who writes wins. Determine, for each pair 
, who has the winning strategy.Note: A power of
is a number of the form 
, where 
is a non-negative integer.
