[/quote]
,
be an acute triangle with incentre 
and 
.
and 
intersect the circumcircle of 
and 
,
and 
such that 
and 
are parallelograms (with 
,
)
be the point of intersection of lines 
and 
.
,
and some point 
on 
.
intersects the circumcircle of 
at 
and the perpendicular from 
intersects the circumcircle of 
at 
.
does not depend on the choice of 
be 
given points in the plane, and let 
be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points 
a non-negative real number 
,
,
.
such that 
and 
are connected by an edge for every 
.
is not fixed and the first selected vertex always has to be 
.![\[
\frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r;
\]](http://latex.artofproblemsolving.com/1/7/9/1795db8e4a509dd465f6ff462093ae75b04de2b4.png)
otherwise, Alice wins. Depending on 
,
for which Bobby has a winning strategy.
,such that 
Prove that : 
Where 
is a positive real number solution of equation : 
,
lie on a line in this order and satisfy 
.
and 
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
,
and 
and 
and 
board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to 
,
to be the largest value in row 
,
.
to be the largest value in column 
.
?
and 
.
(and prove):
be a positive integer and let 
be real numbers.![\[
\left(\sum_{i,j=1}^{n}|x_i-x_j|\right)^2\le\frac{2(n^2-1)}{3}\sum_{i,j=1}^{n}(x_i-x_j)^2.
\]](http://latex.artofproblemsolving.com/8/1/e/81ef4493de1ab4a1ac30124fe291256917760ca1.png)
is an arithmetic sequence.
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
.
be a point on side 
be a point on the extension of 
such that 
.
be the circumcenter of 
,
satisfying 
.
that satisfy the following condition: for all 
and 
relatively prime to 
we have ![\[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]](http://latex.artofproblemsolving.com/e/8/5/e85c5b249ac8e75b45a525b8786414798a4921fe.png)
be distinct prime numbers. Let ![$\alpha_i=\sqrt{a}, \,\mathrm{A}=\mathbb{Q}[\alpha_1,\alpha_2,\hdots,\alpha_b]$](http://latex.artofproblemsolving.com/0/e/0/0e06919fe05d9f17760890bb2aebb9cfa85d54ce.png)
![$[\mathrm{A}:\mathbb{Q}]=2^b$](http://latex.artofproblemsolving.com/b/e/1/be12fe0e1becd8a9dcddd1eece210f2df8ecd026.png)
![$\mathrm{A}=\mathbb{Q}[\gamma];$](http://latex.artofproblemsolving.com/f/6/3/f639e791853133a444708bf47517457379d347e8.png)
and deduce that the minimum polynomial 
of 
over 
has degree 
![$\mathbb{Z}_a[X]$](http://latex.artofproblemsolving.com/e/9/9/e994e6e9083e52f44aef280b4346f6994a9d9f02.png)
into a product of polynomials of degree 
either of degree 
Y by Adventure10
Let 
be an acute-angled triangle such that there exist points 
on side 
, respectively such that the inradii of triangle 
are all equal to 
. If the inradii of triangle 
and are 
and 
, respectively, prove that ![\[ r+r_0=R.\]](//latex.artofproblemsolving.com/6/e/1/6e15d1dbe26ee3d5aef4e6318f5d15c5461d1eda.png)
Soewono, Bandung
be an acute-angled triangle such that there exist points 
on side 
, respectively such that the inradii of triangle 
are all equal to 
. If the inradii of triangle 
and are 
and 
, respectively, prove that ![\[ r+r_0=R.\]](http://latex.artofproblemsolving.com/6/e/1/6e15d1dbe26ee3d5aef4e6318f5d15c5461d1eda.png)
Soewono, Bandung
![\[ r + r_0 = R.\]](http://latex.artofproblemsolving.com/0/f/2/0f2e52a63339989aa2bc46f10e6c38518ef223ce.png)
: 
.
denote the centers of the incircles of 
touch 
at 
touch 
at 
touch 
at 
and 
touch 
at 
Since 
and 
are rectangles and 
it follows that 
and 
are centrally similar through their common incenter 
The similarity coefficient equals the ratio between their perimeters/inradii
from 
into 
yields 
