AoPS Academy
table. It is permissible to reverse the signs of all the numbers in any row or column. Prove that after a number of these operations, we can make the sum of the numbers along each line (row or column) nonnegative.
, prove that 
is 
Find the sum of the digits of
an even number.
is not the product of consecutive integers for infinitely many naturals 
.
, 
. Let 
be the point on 
such that 
. Suppose 
. if 
, Where 
are relatively prime positive integers and 
is a prime number, determine the value of 
.
![\begin{align*}
[(a+1)x-(b-1)y][ax+by]\\
\end{align*}](http://latex.artofproblemsolving.com/b/5/2/b52d93a771b06b5b0a0bebc986de07eda44f25f4.png)
is a polynomial function such that 
then must be greater than 
be an acute, non-isosceles triangle with circumcenter 
, incenter 
and 
tangent to , 
, 
at 
respectively. Suppose that 
cuts 
at 
. Prove that 
bisects segment .
be an acute, non-isosceles triangle with circumcenter , incenter and tangent to , , at respectively. Suppose that cuts at . Prove that bisects segment .
and be midpoint of . Since 
and 
we get 
is cyclic, which we wanted to show.
aand 
Prove that
. Prove that
