AoPS Academy
be a cyclic quadrilateral. Let 
be the midpoint of the arc 
of its circumcircle which does not contain 
or 
. Let the lines 
and 
meet at 
and the lines 
and 
meet at 
. Prove that the lines 
and are parallel.
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
and 
, where 
. She observes that these two graphs and 
axis divides the positive side of the plane (
) into two triangles and a quadrilateral. Find the ratio of sums of two triangles' areas to the area of quadrilateral.
is given, such that could be expressed as difference of two perfect squares (
). Is it possible for 
to be expressed as difference of two perfect squares?
be a cyclic quadrilateral and let the intersection point of lines and 
be 
. Let the points 
and 
be arbitrary points on sides and respectively, which satisfy the conditions
Prove that 
.
square is given, and a quadratic trinomial with a positive leading coefficient is placed in each of its cells. There are 
coefficents in total, and these coefficents are chosen from the set ![$[-66;47]$](http://latex.artofproblemsolving.com/d/d/9/dd92777f4ba30299d3326b285f1f73656aba544b.png)
, and each coefficient is different from each other. Prove that there exists at least one column such that the sum of the six trinomials in that column has a real root.
satisfying the following two conditions:
such that, for all indices 
, 
. and for any index , the number![\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]](http://latex.artofproblemsolving.com/5/4/a/54ac17063104439114f3c7b07de8fd27450ea1eb.png)
is a multiple of .![\[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\]](http://latex.artofproblemsolving.com/5/c/6/5c6894b617b990ec69c62ed1cf7d062d4660af53.png)
where 
are reals.
be a triangle and let 
denote the circumcircle of . The foot of altitude from 
to 
is . The foot of altitudes from to and are 
, respectively. Let 
intersect at 
, and let 
intersect at 
. Prove that is the incenter of triangle 
to the following system:
, which lie on a circle, are marked by distinct reals 
. Let 
be 
good for a on the circle different than , if and only if 
is the greatest number on at least one of the two arcs 
. Let the score of be the number of good points on the circle. Determine the greatest 
such that regardless of the values of , there exists a point with score at least .
be a triangle. 
are points such that 
. 
is the midpoint of , lies on such that 
is the symmedian of . Perpendicular bisector of 
cut 
at . Prove that 
touches two circles with diameters 
and 
be the second points of intersection of 
with 
. be a point in such that 
. and mapping 
to , we conclude that the inverses of 
are the lines 
respectively.
and 
are pairs of inverses. Also, since 
, we see that is the inverse of .
is tangent to the lines and . and are homothetic. But we already know that 
is the reflection of 
over 
.
which is the same as saying 
.
, then 
so 
.
, 
. Then 
, 
for some 
.
to say that 
.
implying that 
.
. And now we just check it:
the centers of 
, 
, the intersection of and 
; 
the tangency points of 
with their common tangent other than :
is the ex-similicenter of and is on their radical axis:
so 
is tangent to hence 
implies 
hence the inversion with center and power 
swaps with 
; 
with 
, the common tangent circle to 
passing through 
; 
with the midpoint of say 
where 
.
but 
hence 
which means 
and 
touches the two circles.
