[/quote]
,
board where we identify the squares with pairs 
where 
and 
denote the row and column number of that square, respectively.
and places a pawn at the bottom left corner (i.e. on 
)
-
or 
if 
and 
if 
.
that Calvin could have picked such that he can make moves so that the pawn covers all the squares on the board without being on any square twice.
such that : 
.
after a correct answer, the score is increased by 
;
.
questions and answers all of them, how many different scores are possible?
and 
with the same length, where 
denotes the correct answer and 
denotes the wrong answer, both result in the same score of 1.
be a circle, and 
are two fixed points on 
on 
and 
denote the feet of the altitudes from 
and 
.
such that 
is tangent to 
Choose a natural number 
such that 
.
are on the board, erase both.
Choose a natural number 
.
are on the board, erase both.
be real numbers satisfying 
Prove that 
and determine all cases of equality.
and circumcentre 
.
cross 
.
,
cross 
at 
and let 
cross 
at 
.
and 
are concurrent on the external bisector of 
.
be a function from the set of integers to the set of positive integers. Suppose that, for any two integers 
and 
is divisible by 
.
,
is divisible by 
.
,
and 
be a pair of isogonal conjugate points. The line 
intersects 
at 
,
intersects 
.
and the circumcircle of 
intersect again at 
(other than 
bisects 
.
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 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.
