[/quote]
,
,
such that 
is a perfect square? (Here 
denotes the integer part of the real number 
?
satisfying
![\[
\text{Find all non-negative integers } (a, b, c, n) \text{ such that}
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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 
,
.
,
,
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 
be a triangle and 
be the foot of the bisector of 
. Let 
and 
be the circumcenters of 
and 
respectively and let 
and 
be the projections of 
and 
through the bisectors of the angles 
and 
respectively. The incircle of touches 
and 
at points 
and 
respectively and the bisectors of angles and meet the perpendicular bisector of 
at points 
and 
respectively. Prove that the five lines 
and 
are all concurrent.
be a triangle and 
be the foot of the bisector of 
. Let 
and 
be the circumcenters of 
and 
respectively and let 
and 
be the projections of 
and 
through the bisectors of the angles 
and 
respectively. The incircle of touches 
and 
at points 
and 
respectively and the bisectors of angles and meet the perpendicular bisector of 
at points 
and 
respectively. Prove that the five lines 
and 
are all concurrent.
