AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be a logarithmic spiral centered at the origin (ie curve satisfying for any point 
on it, line 
makes a fixed angle with the tangent to at ). Let 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.
on the spiral, the polar of wrt. is tangent to the spiral.
+2 w
such that every positive integer appears exactly once and the product of the first 
terms is a perfect 
power ?
be an acute scalene triangle. Let and 
be two distinct interior points of the segment 
such that 
. Suppose that:
and 
are the feet of the perpendiculars from from 
to the lines 
and 
respectively.
and 
are the feet of the perpendiculars from 
to the lines and respectively.
and 
intersect on the line .
be a triangle with incenter 
, and let 
be a point on side . Points and are chosen on lines 
and 
respectively such that 
is a parallelogram. Points 
and 
are chosen on side such that and are the angle bisectors of angles 
and 
respectively. Let 
be the circle tangent to segment 
, the extension of 
past , and the extension of 
past . Prove that is tangent to the circumcircle of triangle .
of the triangle 
which doesn't coincide with 
or 
intersectes the altitudes from 
and 
at 
and 
respectively. Let 
be the reflection of in and 
be the reflection of in 
Prove that the circles 
and 
are tangent.
be positive real numbers such that 
Prove that we have![\[(a+b)(b+c)(c+a)\geq 8.\]](http://latex.artofproblemsolving.com/2/a/6/2a63f19de634fce70ff4ebc02ad13d442a20c378.png)
, points 
are constructed as follows: Points 
and 
lie on the extension of side 
such that 
. Points 
and 
lie on the extension of side 
such that 
. Points 
and 
lie on the extension of side 
such that 
..![\[
\angle ACE + \angle FBD + \angle EAC = 180^\circ.
\]](http://latex.artofproblemsolving.com/c/b/8/cb88e7d6ec129d50bde13e6a08ee16b275f42042.png)
lies inside of scalene triangle with incenter such that 
Lines 
and 
intersect line 
respectively at 
and 
. Line through parallel to 
intersects at and line through parallel to 
intersects at . Prove that triangles 
and are similar.
is the -excenter of the triangle and is the circumcircle of this triangle. 
is the middle of arc of which doesn't contain 
. 
meets at 
. Prove that
, 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 a set with seven elements, where each element is a positive integer not exceeding 
. Prove that there exist positive integers 
(
) such that the equation![\[
x_1 + x_2 + \cdots + x_t = x_{t+1} + x_{t+2} + \cdots + x_m
\]](http://latex.artofproblemsolving.com/7/0/7/707e7e9a0ce45adb307545bbc88c1f0bf90ca0c9.png)
has a solution in the set 
, and 
are all distinct.
