AoPS Academy
if it is not parallel to any of the 
–axis, the 
–axis, or the line 
.
be a given integer. Determine all nonnegative integers 
such that there exist 
distinct lines in the plane satisfying both of the following:
and 
with 
, the point 
lies on at least one of the lines; and of the lines are sunny.
+2 w
and 
be circles with centres 
and 
, respectively, such that the radius of is less than the radius of . Suppose and intersect at two distinct points 
and 
. Line 
intersects at 
and at 
, so that 
lie on in that order. Let 
be the circumcentre of triangle 
. Line 
meets again at 
and meets again at 
. Let 
be the orthocentre of triangle 
. parallel to is tangent to the circumcircle of triangle 
.
+4 w
prime number.
prime number with 
and 
?
be non-negative real numbers such that 
Prove that![\[\frac{10-\sqrt2}{294} \leqslant \frac{(a+b+c)(a^2+b^2+c^2)}{(a+2b+3c)^3} \leqslant 1.\]](http://latex.artofproblemsolving.com/2/f/2/2f2867083d314b8a4825d303b7fbc52e28985a87.png)
When does equality hold?
, and tangents are drawn to this circle parallel to the sides of the triangle. In this way, three smaller triangles are cut off from triangle , each with an incircle of radius 
, respectively. If 
the radius of the incircle of triangle , prove that 
![$ x \in (\frac{\pi}{2}-arcsin(\frac{1}{\sqrt[4]{2}}), arcsin(\frac{1}{\sqrt[4]{2}})) $](http://latex.artofproblemsolving.com/d/5/a/d5a78d72b3a43dd7a1bcf7bf7a1aebe8369d0dcb.png)
, try to prove or disprove that :
that satisfy 
and 
.
.
and 
.
