AoPS Academy
be a logarithmic spiral centered at the origin (ie curve satisfying for any point 
on it, line 
makes a fixed angle with the tangent to at ). Let 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.
on the spiral, the polar of wrt. is tangent to the spiral.
be the set of black cells at the end of the game. Ali and Shayan respectively aim to minimize and maximise the perimeter of the shape by playing optimally. (The perimeter of shape is defined as the total length of the boundary segments between a black and a white cell.), assuming both players play optimally?
called ‘nice’ if and only if 
and 
are both perfect squares. How many positive integers 
such that 
are composites for all nice numbers 
?
decide whether there exist pairwise distinct positive integers 
such that for every 
, 
divides 
.
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
be real numbers such that 
find the largest value of 
such that 
and 
are all perfect square.
be pairwise coprime positive integers with 
being prime and 
. On the segment ![$I = [0, a_1 a_2 \cdots a_n ]$](http://latex.artofproblemsolving.com/a/2/b/a2bce96b048c9b8fc94926db80aba37fd5037b4a.png)
of the real line, mark all integers that are divisible by at least one of the numbers 
. These points split 
into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by .
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
be a regular polygon, and let 
be its set of vertices. Each point in is colored red, white, or blue. A subset of is patriotic if it contains an equal number of points of each color, and a side of is dazzling if its endpoints are of different colors. is patriotic and the number of dazzling edges of is even. Prove that there exists a line, not passing through any point in , dividing into two nonempty patriotic subsets.
be the Fermat point of a 
. Prove that the Euler line of the triangles 
, 
, 
are concurrent and the point of concurrence is 
, the centroid of .
until such that 
is equilateral now, taking as homothetic center the midpoint of 
(3:1) sending X to circumcenter of , P to centroid of and A to centroid of 
that's enough
, 
and 
are the bisectors of 
, 
cutting 
, 
at 
, 
respectively. If 
and 
, find the angles of 
, and are the bisectors of , cutting , at , respectively. If and , find the angles of
, be the midpoint of . Prove that
.
and 
are points on the circumcircle of such that 
is parallel to . Prove that 
is perpendicular to the Simson line of .
be the Fermat point of a . Prove that the Euler line of the triangles , , are concurrent and the point of concurrence is , the centroid of . then the problem is trivial by Liang-Zelich , consider the midpoint of 
, 
, as 
,let the centroid of be 
then we know 
so 
is parallel to 
which is also parallel to 
. now meets the point outside of , as 
such that 
is an equilateral triangle, and now by 
, where 
is the midpoint of we have 
is parallel to so 
lies on therefore it's done
be an acute triangle whose incircle touches sides and at and 
, respectively. Let the angle bisectors of 
and 
meet 
at and 
, respectively, and let the midpoint of be 
. Show that 
is equilateral if and only if 
.
are drawn perpendicular to the sides 
respectively of then prove that
Not been able to solve this even though I know I can do this.
is parallel to , show that tan 
tan 
.
but nvm
be the orthocenter of and let 
be the circumcenter
from and be and 
then just use 
, is the circumcentre and is the orthocentre. Then, prove that 
.
be the antipode and then 
is a rhombus
is parallel to , show that tan tan . but nvm be the orthocenter of and let be the circumcenter from and be and then just use 
now they are the circumcenters of the equilateral triangle formed outside of 
,
,
is concurrent at 
.
,