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.
, 
, and 
Point 
is chosen on the circle with radius , point 
is chosen on the circle with radius , and point 
is chosen on the circle with radius Given that 
, find the maximum and minimum values of 
in terms of , , and
satisfying 
Prove that 
Proposed by Phan Ngoc Chau
, let 
denote the smallest possible value of 
where 
are sets such that 
and 
whenever 
. Determine for each positive integer .
and 
intersect at 
. The common tangent line nearer to 
of the two circles touches 
at 
respectively. Let 
be the symmetric points of with respect to respectively. The circumcircle of triangle 
intersects circles and at points 
respectively which are distinct from . Prove that the circumradii of the triangles 
and 
are equal.
, 
. 
intersects 
at 
. A line 
cuts segment 
at 
respectively. If 
and 
, prove that the diameter of the circumcircle of 
equals 
.
are points on a circle 
, satisfying 
, 
. meets 
at 
. 
is on 
such that 
. Suppose 
intersects again at 
. 
is the reflection of 
wrt 
. Prove that 
lies on the same circle.
in its exterior
we draw the triangles 
so that 
, 
, 
.
and
and be on segment of an acute triangle 
such that 
and 
. Let and 
be the points on 
and 
, respectively, such that is the midpoint of 
and is the midpoint of 
. Prove that the intersection of 
and 
is on the circumference of triangle .
. Each step the pig has probability 
of moving 
unit in the positive direction and probability 
of moving unit in the negative direction. Find the expected number of steps until the yellow pig visits 
.
has a right angle at 
. Let be the point on line 
such that 
and 
lies between and 
. Point 
is chosen so that 
and is the bisector of 
. Point 
is chosen so that 
and 
is the bisector of 
. Let be the midpoint of . Let be the point such that 
is a parallelogram. Prove that 
and 
are concurrent.
such that the number of 
which satisfy the following condition is minimal:
has periodic 
and a point such that 
and 
are tangent to the circumcircle of . Let 
be the reflection of over and 
be the reflection of over 
. If 
is the circumcenter of 
, help Oscar prove that 
is perpendicular to .
in natural numbers m and n.
so as to obtain a vector 
.
，
so 
then its opposite also works, which is 
since they have the same magnitude. If a configuration doesn't work, its opposite doesn't work (likewise). Thus, we must only consider 
Consider the squares of the magnitudes. We must show that one of them is less than or equal to two. The squared magnitude of 
using the dot product and its linearity is 
The squared magnitude of 
is 
Symmetrically, we have 
and 
Adding all four up, we get 
and all the dot products cancel nicely. This means at least one of the squared magnitudes is less than or equal to 
Thus, at least one 
is less than or equal to 
Thus, not only is 
always achievable, so is 
solution is wrong
,
?
so your bound is not tight enough. I think one also must make use of the fact that we are confined to a plane, since the bound is not true in 
.