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such that for all real numbers 
and 
:
then 
then 
be a triangle with circumcentre 
. The points 
and 
are interior points of the sides 
and 
respectively. Let 
and 
be the midpoints of the segments 
and 
. respectively, and let 
be the circle passing through and . Suppose that the line is tangent to the circle . Prove that 
+1 w
be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset 
of such that 
is not divisible by 47 for any 
![$A\in [0,1]^n$](http://latex.artofproblemsolving.com/0/7/e/07eed58d80cf78c7741d61ff142ffffea77a5edf.png)
be an invertible 
matrix with entries between 
and 
. The Frobenius norm of a matrix 
is defined as![\[\Vert X \Vert_F = \sqrt{\mathrm{tr}(XX^T)} = \left(\sum_{i,j=1}^n X_{ij}^2\right)^{1/2}.\]](http://latex.artofproblemsolving.com/a/2/a/a2ac3606933c81aebd6e9b26af765e7202a23aec.png)
Prove that![\[\Vert A^{-1} \Vert_F \ge \frac{2n}{n+1}\]](http://latex.artofproblemsolving.com/9/6/b/96b79f60d6d54816ac3c7e7cd7b6b95f823d4b75.png)
for (a) 
and (b) sufficiently large odd 
. is large enough, in every square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by 
.
is large enough, in every square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by .
, be the matrix corresponding to entries in the square.
, where 
.
over a sub-square, 
, equals the sum 
.
be a colouring of the lattice. If is sufficiently large, then by Gallai's theorem there exists 
monochromatic lattice points that are vertices of a square. These correspond to four entries of 
such that 
. This is exactly what we wanted to prove.
