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 positive real 
, define 
.
for all integer 
.
for all integer and real 
., there exists some positive 
such that 
for all positive integer .
and positive real 
such that 
, we have 
.
+1 w
on one side and 
on the other side. Bob arranges 
coins in order from left to right and repeatedly performs an operation: Bob selects the coin that is not the leftmost and rightmost coin with its neighbours show to different faces, then flip that coin. For example, if 
with the initial configuration 
, Bob can only flip the third coin and make the configuration 
. Let 
and 
be any two configurations. Prove that if he can turn into by using an order of allowed operations, then he could complete the operation in at most 
times.
, 
, 
be positive real numbers such that 
. Prove that
such that the sequence of non-negative integers 
with 
and 
is bounded. and , define 
to be a unique integer 
such that divides 
.
be an acute-angled triangle, and let 
and 
be two points on its side 
. Construct a point 
in such a way that the convex quadrilateral 
is cyclic, 
, and and lie on opposite sides of line 
. Construct a point 
in such a way that the convex quadrilateral 
is cyclic, 
, and and lie on opposite sides of line 
. Prove that the points , , , and lie on a circle.
satisfying [list] [*] For each positive integers 
, there exists exactly 
's positive integers such that 
. [*] 
for every positive integers and 
. [/list]
+1 w
be a sequence of positive real numbers with the property that
for all positive integers . Show that 
.
be a triangle such that 
with 
and are circumcenter and orthocenter respectively. Let be an intersection of 
and 
.
be a midpoint of 
and a circle with diameter meets the circumcircle of 
again at 
.
be a midpoint of 
and circle with diameter meets the circumcircle of again at 
.
and 
intersect at a point.
and be positive integers. Prove that there exists infinitely many positive integer such that 
is not a perfect square.
be positive integers and 
be a polynomial with non-negative integer coefficients. Prove that there exists a positive integer 
such that divides 
.
be an equilateral triangle. Let 
be a point on the side 
. Let the perpendicular bisector of 
meets 
and 
at point 
and 
respectively. Suppose that the circumcircle of 
meets at 
. Prove that 
.
be an acute scalene triangle. Let be the foot of altitude from 
to . Let be a point on the segment such that 
and be a point on the segment such that 
. Construct point on the line 
such that 
. Similarly, construct point on the line 
such that 
.
is on the internal angle bisector of 
.
be the foot of the altitude from the vertex 
in the acute-angled 
. Let be the midpoint of side , and be the circumcentre of . Line 
meets line 
at . Prove that the points 
, and lie on a circle.
is right-angled and is the midpoint of , we see 
. Hence, 
and 
are isosceles. We perform a simple angle chase.
, then 
. Also, by inscribed angles,![\[
\angle BOC = 2 \cdot \angle BAC = 2x,
\]](http://latex.artofproblemsolving.com/7/9/9/799b5b58000a6f785866d9e8c11497dc010d6a5c.png)
and since 
, we have![\[
\angle BCO = 90^\circ - x.
\]](http://latex.artofproblemsolving.com/0/f/f/0ff65392a68cbbe475c4b4a83f7547d8cc7071d0.png)
Therefore,![\[
\angle EB_1B = \angle OCB = 90^\circ - x,
\]](http://latex.artofproblemsolving.com/0/f/5/0f5ec2fee5f7a5f4c56d6821a66fb9ab27e0c054.png)
finishing our proof.
which satisfy:
