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 
.
)
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
+1 w
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)
be a parallelogram and points 
be on its exterior. If triangles 
and 
are similar, i.e. 
, prove that triangle 
is similar to these two triangles.
be a triangle and let be a point not lying on any of the three lines 
, 
, or 
. Distinct points 
, 
, and 
lie on lines , 
, and , respectively, such that 
and 
. Show that there exists a point 
on the circumcircle of 
such that 
is similar to 
.![\[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\]](http://latex.artofproblemsolving.com/5/c/6/5c6894b617b990ec69c62ed1cf7d062d4660af53.png)
where 
are reals.
is inscribed in circle 
. The interior angle bisector of angle 
intersects side and at and 
(other than ), respectively. Let be the midpoint of side . The circumcircle of triangle 
intersects sides and again at and (other than ), respectively. Let be the midpoint of segment 
, and let be the foot of the perpendicular from to line 
. Prove that line 
is tangent to the circumcircle of triangle 
.
, then we can use QF to find 
.
is a 36-72-72 triangle. Thus, is the foot of the angle bisector from 
, and 
, 
. We are done.
and 
.
and passes the point 
,
.