such that ![$1+[\sqrt{2n}]~$](http://latex.artofproblemsolving.com/e/6/a/e6aa1bf5d761758196027add39f2e3cd77db1c1c.png)
divides 
.
, ![$[x]$](http://latex.artofproblemsolving.com/b/c/e/bceb7b14e55d33a8bca29b7863ad3cdae95afce4.png)
denotes the largest integer not exceeding 
x
be a scalene triangle with circumcenter 
,
,
.
,
,
at points 
,
,
respectively. Let 
be the projection of 
.
intersects the circumcircle of 
.
and 
intersect 
and 
respectively. Prove that the lines 
,
,
are concurrent.
of integers such that
and all triples 
are integers.
.
Let 
.
be the number of positive integers 
and 
divides
where 
denotes the number of positive integers that divide 
and 
.
and 
then find the value of 
and 
Show that
Let 
Show that
and 
are in the coordinate plane such that 
.
denote the locus of all points 
in the coordinate plane satisfying 
,
be the midpoint of 
.
and 
are on 
and 
.
can be expressed in the form 
,
and 
.
.
Prove that
Where 
are positive integers and 
is a non-square positive integer. If 
denotes the n-th set of solution to the equation with 
Then, prove that 
Y by
Determine the largest two-digit number 
with the following property:
for any six-digit number
number is a divisor of the number if and only if the number is a divisor of the corresponding three-digit number 
.
Note The numbers
and 
need not be different.
with the following property:for any six-digit number
number is a divisor of the number if and only if the number is a divisor of the corresponding three-digit number 
.Note The numbers
and 
need not be different.
