.
on 
respectively. Circumcircle 
of triangle 
intersects 
.
intersects 
at 
.
cuts 
respectively. Construct parallelogram 
.
are collinear.
and 
are complex numbers such that
a grid of 9*9, we choose a little square in 
complex numbers with 
such that
Find the value of 
.
is a strictly increasing sequence of positive integers such that the sub-sequences ![\[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\]](http://latex.artofproblemsolving.com/2/6/8/2686c6e4c1110bce322dd876aba37d5273bffe05.png)
are both arithmetic progressions. Prove that the sequence 
is itself an arithmetic progression.
denote the set of natural numbers and let 
denote 
ordered pairs in 
.
distinct elements in the set 
for 
whose product is a perfect cube.
Where 
are positive integers and 
is a non-square positive integer. If 
denotes the n-th set of solution to the equation with 
Then, prove that 
and 
Show that
Let 
Show that
have real roots and real coefficients with 
.
?
,
.
have roots 
.
(I won't be proving this here). This new information allows us to utilize the mean inequality chain as follows:
Similarly, applying AM-HM yields 
.
.
,
.
.
.
such that:
be a prime number. How many different pairs 
satisfy the equation:
Y by chessgocube
An acute-angled triangle 
and its Euler circle are given. Two different circles are drawn through vertices 
and 
, tangent to Euler's circle at points 
and 
. Points 
are obtained by similar constructions. Prove that three straight lines 
, 
, 
intersect at one point belonging to the Euler line.
(D. Soskin)
and its Euler circle are given. Two different circles are drawn through vertices 
and 
, tangent to Euler's circle at points 
and 
. Points 
are obtained by similar constructions. Prove that three straight lines 
, 
, 
intersect at one point belonging to the Euler line.(D. Soskin)
to the opposite sides be 
.
are cyclic, so 
is on the radical axis of 
and the 
must be the polar of 
similarly. 
on the radical axis of 
