AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
points in 
can be partitioned into two sets whose convex hulls intersect.
and 
and 
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer 
such that 
is called the period of 
. Determine the total number of points in 
of period 
.
)
and 
are two sequences between 
and 
and they satisfy 
, prove that 
, and determine when does equality hold
be the positive number satisfied 
Find the minimum of 
, we have 
When does the equality hold?
. If 
, then prove the following: 
people be arranged in a circle, numbered clockwise from to . These people are eliminated one by one in order, until only one person remains. The elimination follows this rule: among the remaining people, start counting clockwise from the person with the smallest number, and eliminate the -th person in that count. Then, among the remaining people, start counting again from the person with the smallest number and eliminate the -th person. Repeat this process until only one person remains. Let 
denote the number of the last remaining person.
, people are eliminated in the following order: 
. Thus, 
. It is known that 
under certain conditions. Prove that the necessary and sufficient condition for this is that both 
and 
are prime numbers.
consecutive even numbers, such that the sum of their squares divides the square of their product.
. Solve the equation: 
be a triangle and let 
be its circumcenter and 
its incenter.
be the radical center of its three mixtilinears and let 
be the isogonal conjugate of .
be the Gergonne point of the triangle .
is parallel with line 
.
be a positive integer. Prove that there exist three permutations 
, 
, and 
of 
such that ![\[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\]](http://latex.artofproblemsolving.com/9/8/0/98097fb721722b9103ef65d72322956877e262cb.png)
for every 
.
at 
and 
meet at 
. The circumcircle of 
meets sides 
and 
again at 
and 
respectively. Let be the circumcentre of . Show that 
is perpendicular to 
.
is defined as the intersection of the tangents. We can replace the circumcircle of with any circle that passes through and . Indeed, we know that 
, and since 
is cyclic, we have that 
, from which the result follows.
by reflecting 
about its 
angle bisector and taking a homothety at 
. Now since the circumcentre and orthocentre are isogonal conjugates, has to lie on the altitude of , and therefore we are done. 
be the orthocenter of .
is concyclic 
.
, such that 
goes to . Under this transformation, line 
goes to line because they are isogonal in . Since 
, we have 
.
lies on the circumcircle of 
, we know that
so
we know that
hence we are done 
hence lies on perpendicular bisector of and lies on perpendicular bisector of moreover also lies on these lines
is the orthocentre of hence 
is perpendicular to 
. Prove that
.
is the inverse of function 
and thus, they must intersect that 
axis (as its clearly not self-inverse function). Thus: 
is the only solution.
is the 
are concyclic. So 
.
,
since 
are concyclic. Therefore, 
.
(because CFEB is cyclic)
