[/quote]
,
has a right angle at 
.
be the point on line 
such that 
and 
lies between 
.
is chosen so that 
and 
is the bisector of 
.
is chosen so that 
and 
is the bisector of 
.
be the midpoint of 
be the point such that 
is a parallelogram. Prove that 
and 
are concurrent.
and 
Prove that
Let 
Prove that
.
the number of changes of signs in the
Then prove that
.
epic if for any four different positive integers 
,
such that 
can be partitioned into 
be positive real numbers such that 
.
![\[\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1\]](http://latex.artofproblemsolving.com/f/a/6/fa6c46c12e7e6bf0b04f6f9a910df6c2b452fe6a.png)
for all real numbers 
,
,
,
,
.
be a prime and suppose 
is prime. Prove that 
is prim
are positive real numbers such that 
prove that
girls on one side of a tennis table, and a queue of 
to 
grid. We color the squares with three colors: blue, red, and yellow. Two squares are defined as opposite if they share a vertex but not an edge. A valid coloring is a coloring such that no two squares that are red and blue are opposite each other. Count the number of valid colorings.
,
and 
.
be the intersection of 
and 
and 
and 
.
is a cyclic quadrilateral prove that 
,
Y by
Suppose that 
consecutive positive integers were written on the board, where 
. Then some 
of the written numbers were erased, and it turned out that any two of the remaining numbers are coprime. Find the largest possible value of .
consecutive positive integers were written on the board, where 
. Then some 
of the written numbers were erased, and it turned out that any two of the remaining numbers are coprime. Find the largest possible value of .This post has been edited 1 time. Last edited by BR1F1SZ, Yesterday at 12:59 AM
then there are at least 
even numbers, and after erasing, there are at least 
even numbers, which leads to contradiction
or 
then there are 
even numbers, and 
numbers, which are divisible by 
and at least 
and for 
we can choose numbers 
and erase non-prime numbers except 
