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and 
such that 
and 
is a prime number
, let 
be the midpoint of side 
. Let the line perpendicular to 
at point 
intersect 
at 
. If 
is tangent to 
at 
, find 
.
of side length 
lies in the union of five equilateral triangles of side length 
, show that there exist four equilateral triangles of side length whose union contains .
is a polynomial of degree 
such that
.
write all its divisors in increasing order: 
. such that 
.
is called good if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
is a quadratic residue modulo 
for every 
is considered a quadratic residue modulo if there exists an integer 
such that 
.
be a triangle. Let 
be a circle passing through intersecting at 
and at 
. Let 
be the intersection of 
and . Further, let and be the intersections of 
and 
with the tangent to 
at 
. Now, let 
be the second intersection of 
and . Then, prove that , , , 
and are concyclic.
, 
is the midpoint of 
. 
is an arbitrary point on 
. Let 
be the intersection of 
and 
. The line 
intersects with the circumcircle of 
, for the second time at 
. 
is the circumcenter of triangle 
. Prove that 
.
) and you also need to be quite good at algebra. You should start with basic euclidean though and perhaps learn inversion first before trying bashing, you will build the fundamentals you need that way.
be real numbers such that 
.
