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it can draw the line which forms the interior angle bisector of 
.
in the plane . Prove that Euclid can use the splitter several times to draw the centre of circle passing through 
and 
.
, 
satisfying 
, what is the maximum possible value of 
?
and 
such that![\[
1^n + 2^n + 3^n + \cdots + n^n = x!
\]](http://latex.artofproblemsolving.com/6/4/b/64b5f5919fd3710cc15e96724918eae216c8e4f0.png)
and 
. Prove that
Equality holds when 
Equality holds when 
Equality holds when 
and 
Prove that
Let and 
Prove that
Let and 
Prove that
Let and 
Prove that
with 
. Denote the orthocenter of 
as 
, the intersection of 
and 
as 
. Compute the length of 
.
with 
and 
. Let 
be a point on 
such that 
. Suppose that the angle bisector of 
is tangent to the circle with diameter 
and say it intersects at point 
. Find the length of 
.
, and be four different points in the plane. Three of the line segments 
,
have length 
. The other three have length 
, where 
. Determine all possible values of the quotient 
.
, and be four different points in the plane. Three of the line segments , have length . The other three have length , where . Determine all possible values of the quotient .
is equilateral. Let is equilateral. Then -center of so 
,
and 
is equilateral. Let 
. Then 
or 
equal , let it is and 
.
and so 
: 
Prove that
![\[
\frac{2a^2}{(b-c)^2} + \frac{2b^2}{(a-c)^2} + \frac{2c^2}{(b-a)^2} \geq 2 + \sum \left| \frac{(a+b-c)(b+c-a)}{(c-a)^2} \right|
\]](http://latex.artofproblemsolving.com/4/8/1/4813b0f718b56d880792297a5fe6141ad2e1f2ab.png)
and 
.
