be triangle with circumcenter 
and orthocenter 
.
intersect 
respectively. Lines through 
intersects 
at 
respectively. Point 
such that 
is a parallelogram. Prove that lines 
and 
are concurrent at a point on 
.
be the point such that the quadrilateral 
is a parallelogram. Let 
be the point on the line 
such that 
bisects 
.
at 
and 
.
.
satisfying
Prove
and 
be any point in the plane. Prove that 
has at most one solution continuous for 
)
is a fixed real number. (A. 7)
converge?
,
where 
is the identity matrix. If for some matrix 
is identity, does it follow that 
be real numbers. Find the ranges of 
converges.
are group. If 
,
or 
.
are complex numbers such that
and if 
denotes the sum of all products of these 
numbers taken 
at a time, prove that
whenever the denominator of the left-hand side is different from 
.
for any positive reals satifying 
?
are real number satisfying
show that the equation 
has at least one real root.
Y by
Let 
be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of 
. It is known that the graph of contains exactly 
chords, one of which has length 2025. Find the minimum possible value of .
be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of 
. It is known that the graph of contains exactly 
chords, one of which has length 2025. Find the minimum possible value of .
?
