IMO 2009 P2, but in space
by Miquel-point, May 14, 2025, 6:35 PM
Let 
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 
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)
,
and a point 
in its interior, not coinciding with the center of 
,
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 
and 
are not relatively primes.
let 
and 
denote the feet of the altitudes from 
and 
,
intersect circumcircle 
.
intersect circumcircle 
.
and 
passes through the orthocenter of triangle 
of integers such that
and all triples 
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 
.
is a simple root of ![\[ \sum_{k=1}^m \frac{1}{w_k - u_0} = 2\sum_{k = 1}^n\frac{1}{u_k - u_0}.\]](http://latex.artofproblemsolving.com/6/f/3/6f36b138d97a24d1941d21f7ffbc11530ca2adde.png)
September 2012
May 2012