and 
.
identical cars on a circular track. Among all of them, they have just enough gas for one car to complete a lap. Show that there is a car that can complete a lap by collecting gas from the other cars on its way around
,
be its circumcenter. Let 
and 
be orthocenters of triangle 
and 
,
be circumcenter of triangle 
.
and circle 
intersect at 
for the second time, prove that 
.
is a polynomial with real coefficients such that![\[ P(x)^2 - 1 = 4 P(x^2 - 4x + 1). \]](http://latex.artofproblemsolving.com/b/4/2/b425a938a1dbbac25e0e0793b30c9e58845d45ab.png)
Find 
and 
be altitudes in an acute triangle 
.
meets the circumcircle of 
and 
.
and 
meet at 
.
and 
and the line 
concur.
be a diameter of a circle 
,
be the tangent at 
.
be a fixed, positive real, and consider all pairs of points 
and 
lying on 
.
and 
intersect 
and 
,
pass through a common point.
such that
for all 
satisfying 
,
,
,
and 
intersect the altitude from 
to 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of 
is the second intersection of circumcircles of triangles 
and 
.
are collinear.
Y by buratinogigle
Inside triangle 
, point 
is marked. Point 
is on segment 
, and point 
is on segment 
such that the circumcircles of triangles 
and 
are tangent to line 
. Lines are drawn through points 
and 
passing through the center of the circumcircle of triangle 
, and through points and passing through the center of the circumcircle of triangle 
. Prove that there exists a circle tangent to all four drawn lines.
, point 
is marked. Point 
is on segment 
, and point 
is on segment 
such that the circumcircles of triangles 
and 
are tangent to line 
. Lines are drawn through points 
and 
passing through the center of the circumcircle of triangle 
, and through points and passing through the center of the circumcircle of triangle 
. Prove that there exists a circle tangent to all four drawn lines.
be the centers of 
respectively (last circle exists from invert at 
so dw) we claim that the quadrilateral asked for has as incenter 
,
.![\[ \measuredangle O_2RC=\measuredangle PRC-\measuredangle PRO_2=\measuredangle APC+90+\measuredangle RQP=\measuredangle APC+90+\measuredangle RQB+\measuredangle BQP=\measuredangle BPC+90+\measuredangle ACB=\measuredangle BCO_1+\measuredangle ACB=\measuredangle RCO_1 \]](http://latex.artofproblemsolving.com/d/7/2/d72bd2b569e03b5fbc524303576c6f4721b28094.png)
And this shows that the perpendicular bisector of 
becomes an angle bisector so you do the same to get that so does 
and from here we are obviously done 
.
.
maps 
to 
,
are lie on a circle, assume centered at 
and 
,
intersects 
(other than 
)
and 
is an isosceles triangle, which implies 
.
