[/quote]
,
be a positive integer. Consider the following equation: ![\[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\]](http://latex.artofproblemsolving.com/8/2/9/8299b3ceb1650418c0a373536cb1d2931900efd9.png)
a) For 
,
.
real solutions.
, where 
externally bisect 
and let 
be the line passing through 
which is parallel to line 
.
at point 
,
at point 
.
meet line 
at point 
and the internal bisector of 
at point 
.
,
and 
.
then prove
be an acute-angled triangle, with circumcenter 
,
and orthocenter 
be a point on 
such that 
.
and 
similarly.
,
be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer 
(not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are 
and 
respectively, we replace them with 
and 
,
.
be the set of all positive integers. Determine all subsets 
of 
such that
Y by Adventure10, Mango247, Mango247
Let 
be a square of side 
. On side 
consider points 
and 
such that 
and 
. If the lines 
and 
intersect at point 
, calculate the area of the triangle 
in terms of .
be a square of side 
. On side 
consider points 
and 
such that 
and 
. If the lines 
and 
intersect at point 
, calculate the area of the triangle 
in terms of .This post has been edited 2 times. Last edited by parmenides51, Jul 15, 2019, 10:42 AM
Reason: name typo
Reason: name typo
.
.
such that 
.
.![$[HZE]=\frac{25a^2}{168}$](http://latex.artofproblemsolving.com/b/a/f/baf0fb3827c570ee24713ea7aa9b6f40aa272dcf.png)
.![$\frac{[HZE]}{[BHC]}=(\frac{5}{12})^2\implies [BHC]=\frac{6a^2}{7}$](http://latex.artofproblemsolving.com/2/e/0/2e0ecdd82402a674d5c1dcdeff516f4952106a45.png)
.
