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.
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 
and 
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 
.
, no matter how its elements are partitioned into two subsets, at least one of the subsets must contain three numbers 
(
is allowed) such that 
. Find the minimal .
and a point , let 
be the cevian triangle of with respect to . Let 
be the orthocenter of , and denote the isotomic conjugate of 
with respect to by 
, respectively. Let the centroid of be 
, and denote the isogonal conjugate of with respect to by 
. Prove that
or in brief
be the circumcenter and the orthocenter of an acute triangle . Prove that the area of one of the triangles 
, 
and 
is equal to the sum of the areas of the other two.
line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands 
times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time. is odd. is even.
such that a positive integer power of 
equals to a positive integer power of 
.
can be written in the form
where 
for some non-negative 
such that
for every 
.
to the following system:
![\[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\]](http://latex.artofproblemsolving.com/9/2/a/92a524bbdca92e9dd4091275714bf346bad96cd7.png)
+1 w
?
?
?
where 
divisible by 
prime numbers prime
is a prime number, show that 
is composite.
or 
.
,
,
which is impossible, so 
.
.
such that 
is a perfect cube.
,
,
.
where p is a prime,then 
where 
we have 
,
If for an odd prime 
we have 
,
which is also impossible(it is also clear that k=0 or k=1).Then 
.
then 
,
.
or 
It is trivial that 
.
.
has residue 
mod 
,
.
,
.
doesn't divide 
,
so 
,
.
take 
take 
take 
a succession of positive integers such that for every couple of positive integers 
we have 
. Prove that there exists a positive integer 
such that 
.
is obvious. Let 
be called a good pair if 
for some 
.
are 
distinct pairwise good integers.
is a good pair iff 
.
,
then let 
,
to both sides gives 
so the claim is proven.
,
for 
and 
.
are 
pairwise good integers. First we show that 
are pairwise good. This follows from 
by induction and our definition of 
.
is a good pair. Note that 
as desired, so we have 
.
and 
such that 
and 
we have![\[
\text{gcd}(x+y,mn)>1
\]](http://latex.artofproblemsolving.com/0/e/3/0e3bae9e964367fac00f7f728b1db4fecf514ba3.png)
be the subset of primes dividing 
the subset of primes dividing 
such the two are disjoint. Further, take 
to be the set of overlap in 
prime factors. If none exist take it to be 
.
hence 
,
.
such for any 
:
.
and 
are coprime and 
,
-
.
.
t
.