AoPS Academy
be a convex quadrilateral, and 
, 
, and 
be the midpoints of diagonals 
and 
, and side 
, respectively. Also, suppose that 
and that 
, 
. Calculate the perimeter of triangle 
.
or 
tile.
be an integer. Dominoes are placed on an 
board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some 
such that each row and each column has a value of 
. Prove that a balanced configuration exists for every , and find the minimum number of dominoes needed in such a configuration.
be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer 
such that 
. such that 
and a constant 
, it is given that 
and 
are also friendly polynomials. Prove that 
.
and a constant , it is given that and are also friendly polynomials. Prove that .
and 
are distinct, and without loss of generality, let have degree d higher than . Every root of 
and 
is a root of 
, and and cannot share any roots. If has a root 
with multiplicity 
, then has multiplicity 
in 
. Therefore, the number of distinct roots of equals 
minus the number of roots of 
where 
. Then the number of distinct roots of equals 
minus the roots of 
with or 
. The value is at least 
, creating a contradiction.
