of positive integers for which ![$a + [a, b] = b + [b, c] = c + [c, a]$](http://latex.artofproblemsolving.com/a/1/0/a10eb969d7e061a14a11cfeccdc043f4fbbfbc1b.png)
.![$[a, b]$](http://latex.artofproblemsolving.com/d/a/2/da2e551d2ca2155b8d8f4935d2e9757722c9bab6.png)
denotes the least common multiple of integers 
.
and 
.
chess board. The allowed move is to add an integer 
to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.
be positive integers such that ![$\sqrt[n]{a}+\sqrt[n]{b}$](http://latex.artofproblemsolving.com/d/a/a/daaf95cd3381060559529aa4a318e97b727eac31.png)
is an integer. Prove that 
are both the 
t
positive integers.
Y by Adventure10 and 2 other users
In a trapezoid 
with a base 
, point 
is the orthogonal projection of 
on 
, and 
is the point on 
such that 
is perpendicular to . Let 
be the circumcenter of triangle 
. Suppose that the lines 
and 
have a common point. Prove that is a parallelogram.
with a base 
, point 
is the orthogonal projection of 
on 
, and 
is the point on 
such that 
is perpendicular to . Let 
be the circumcenter of triangle 
. Suppose that the lines 
and 
have a common point. Prove that is a parallelogram.
be the circumcircle of 
be the intersection of 
,
.
be the intersection of 
and 
is a diameter of 
but we knew that 
lie on a line,so 
.
so its sufficient to show that 
.
we get that 
so in triangle 
we get that 
.
s
we get the result...
,
,
so 
thus 
is cyclic.
