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 
.
. Then 
is cyclic and 
.
. Since 
and 
we have that 
is the diameter of the circumscribed circle of 
. Then 
.
is cyclic, which implies that, 
Also, 
, Now since, 
. 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.
be a square and 
points on sides 
respectively such that 
.
is the midpoint of 
where 
are points of intersection of 
with the lines 
.
it's said 
?
and 
are cyclic. angle bash some more. and then prove 
is cyclic, upon noticing 
is diameter. 
is centre hence 
oops sorry. i thought ur soln had an error. but it was actually a typo. sorry 
