AoPS Academy
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, 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.
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples of real numbers, satisfying the equations, are such that 
are integers.
be a rational function with complex coefficients whose denominator does not have multiple roots. Let 
be the complex roots of 
and 
be the roots of 
. Suppose that 
is a simple root of . Prove that![\[ \sum_{k=1}^m \frac{1}{w_k - u_0} = 2\sum_{k = 1}^n\frac{1}{u_k - u_0}.\]](http://latex.artofproblemsolving.com/6/f/3/6f36b138d97a24d1941d21f7ffbc11530ca2adde.png)
be a tetrahedron with circumcenter 
. Suppose that the points 
and 
are interior points of the edges 
and 
, respectively. Let 
and 
be the centroids of the triangles 
, 
and 
, respectively. Prove that if the plane is tangent to the sphere 
then 
where 
is the number of vertices. What is the length of the shortest possible cycle in the graph?
in this order. Prove that![\[a_1a_3+a_3a_5+a_5a_7+a_7a_9+a_9a_1=a_2a_4+a_4a_6+a_6a_8+a_8a_{10}+a_{10}a_2.\]](http://latex.artofproblemsolving.com/9/e/e/9ee73bbdc4b4f2cad2eb3fcfb3dbdf76b6200b4d.png)
, given a circle 
and a point 
in its interior, not coinciding with the center of . Call a point of space, not lying on , a proper projection center if there exists a plane 
, not passing through , such that, by projecting the points of from to , the projection of is also a circle, and its center is the projection of . Show that the proper projection centers lie on a circle.
vertices are coloured in two colours. Prove that the number of cycles formed by four edges of the same colour is more than 
.
, 
, and 
there exists infinitely many positive integers for which 
and 
are not relatively primes.
let 
and 
denote the feet of the altitudes from 
and 
, respectively. Let line 
intersect circumcircle at points 
. Similarly, let line 
intersect circumcircle at points 
. Prove that the radical axis of circles 
and 
passes through the orthocenter of triangle
which gives B