AoPS Academy
is situated inside the parallelogram 
such that 
. Prove that 
.
inscribed in 
with 
being the midpoint of 
. The tangents at 
of intersect at 
. Let 
be the projection of onto 
. On the perpendicular bisector of , take a point 
that is not on and different from M. Circle 
intersects 
at 
. Lines 
and 
intersect at 
. Prove that 
is an isosceles triangle.
be a triangle with circumcircle 
. Arbitrary points 
on 
respectively. Circumcircle 
of triangle intersects at 
. 
intersects 
at 
. 
cuts and at 
respectively. Construct parallelogram 
. Prove that 
are collinear.
a grid of 9*9, we choose a little square in of this grid three times, we can choose three times the same.
complex numbers with 
such that
Find the value of 
.
denote the set of natural numbers and let 
denote 
ordered pairs in 
. Prove that there exist 
distinct elements in the set 
for 
whose product is a perfect cube.
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 
have real roots and real coefficients with 
. What is the maximum value of 
?
, all the roots added will give 
. I also learned that if every single root, multiplies every single root, it will give 
. I also learned that if all the roots are multiplied, it will give 
. Is this right? And is there any purpose for these equations?
such that:
be a natural number and 
be a prime number. How many different pairs 
satisfy the equation:
, 
is parallel to one of the sides of a polygon. We have 
diagonals, and only 
sides, which means that there is a side which is parallel to at least 
diagonals, thus these digonals are pairwise parallel, which means that they cannot share a common end. So these diagonals give us 
different enpoints. Together with our chosen side, this gives us different endpionts (obvoiusly these diagonals cannot share a common endpoint with our side), which are all vertex of our polygon. Thus one of our diagonals must be a side of the polygon ( impossible because of assumption about convexity)
and 
are complex numbers such that
and 
Show that
Let 
Show that
,
.
have roots 
.
(I won't be proving this here). This new information allows us to utilize the mean inequality chain as follows:
Similarly, applying AM-HM yields 
.
.
be a convex polygon with an even number of corners. Prove that there exists a diagonal 
which is not parallel to any side of the polygon.
