[/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 
.
,
Find the range of values that can be taken by :
are positive reals.
there are elements 
such that 
.
would be divisible by 
for any integer values of 
,
is not a factor of 
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 
,
candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it:
She eats a candy of type 
,
followed by one candy type 
(we consider type 
to be type 1, and type 0 to be type 
She chooses two consecutive candies which are the same type, and eats them.
such that 
has a prime divisor greater than 
.
and 
Prove that
be an interior point of the convex quadrilateral 
.
and 
on the outside of the quadrilateral such that the similarities 
and 
hold. Let 
and 
and 
,
is cyclic, then![\[EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.\]](http://latex.artofproblemsolving.com/0/8/e/08e13ecbf07fb1e8950fa34e40cb7f08a8c5a7e1.png)
,
be all the positive integers smaller than 
such that
for all 
.
and 
are arranged so that 
lies on the segment 
and 
.
is the midpoint of 
.
.
be reals such that 
Prove that
Y by Adventure10, Mango247
In a circle 
with centre 
and radius 
, let 
, 
be two circles with centres 
, 
and radii 
, 
respectively, so that each circle 
is internally tangent to at 
and so that , are externally tangent to each other at 
.
Prove that the three lines
, 
, and 
are concurrent.
with centre 
and radius 
, let 
, 
be two circles with centres 
, 
and radii 
, 
respectively, so that each circle 
is internally tangent to at 
and so that , are externally tangent to each other at 
.Prove that the three lines
, 
, and 
are concurrent.
to 
,
.
to 
balance at 
to 
,
balance at 
at 
.
now balance at 
,
in the triangle 
are concurrent.
,
,
in 
,
so 
are on same line 
a
are on same line.
is perpendicular to line that tangent to circl
is perpendicular to line that tangent to circl
are on same line 
!
and 
and 
AND 
we must proof:
a
so the problem is proof!
.![\[\frac{OA_1}{A_1O_1} \cdot \frac{O_1A}{AO_2} \cdot \frac{O_2A_2}{A_2O} = 1.\]](http://latex.artofproblemsolving.com/5/8/f/58fe3d80b59713a430f2bccd0d008e182f16eb4c.png)
![\[\frac{r}{r_1} \cdot \frac{r_1}{r_2} \cdot \frac{r_2}{r} = 1,\]](http://latex.artofproblemsolving.com/6/d/9/6d92534ae3bab68d714a138855a202469eb1b2e7.png)
