[/quote]
,
are non-negative real numbers such that 
Prove that![\[a\sqrt{72b+2ca(81+11b)} + b\sqrt{72a+2bc(81+11a)} + c\sqrt{72c+2ab(81+11c)} \leqslant 48.\]](http://latex.artofproblemsolving.com/6/0/f/60f787cdb45d8c34bfe570ff2e1291ee2a17a411.png)
or 
satisfies the following condition.
,![$\sum_{k=1}^{n}\frac{1}{2}\left(1 - (-1)^{\left[\frac{n}{k}\right]}\right)a_k=1$](http://latex.artofproblemsolving.com/3/1/0/3109e3f024dc59c2cdad60efa270494d6095c330.png)
holds.
,
.
.
.
be real numbers such that 
and 
.
.
are relatively prime positive integers, 
is irreducible in the complex numbers. (A polynomial 
is irreducible if there do not exist nonconstant polynomials 
and 
such that 
for all 
.
be a triangle with an obtuse angle at 
and 
be the intersections of the external bisector of angle 
respectively. Let 
and 
be the points on the segments 
and 
respectively such that 
and 
.
lie on a circle.
.
such that
for all 
.
and 
Prove that
Let 
Prove that
be all the positive integers smaller than 
such that
for all 
.
table can be filled with one of the letters 
and 
in such a way that:
,
to 
with 
.
,
diagonals of two types. A diagonal of first type consists all cells 
is a constant, and the diagonal of this second type consists all cells 
is constant.
be integers. For each permutation 
of 
,
.
of 
is a divisor of 
.
.
students take a test. For any two question in a test, there are at least 
students who solved exactly one of those two questions. Prove that there are no more than 
questions in this test. Showing the equality case is not needed.
Y by Adventure10, Mango247
Consider a pyramid whose base is an equilateral triangle 
and whose other faces are triangles isosceles, right at the common vertex 
. An ant leaves the vertex 
arrives at a point 
of the 
edge, from there goes to a point 
of the edge 
and returns to point . If the path you made is minimal, how much is the angle 
?
and whose other faces are triangles isosceles, right at the common vertex 
. An ant leaves the vertex 
arrives at a point 
of the 
edge, from there goes to a point 
of the edge 
and returns to point . If the path you made is minimal, how much is the angle 
?
just unfold it
