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.
is written on the board. A move consists of adding either , 
or 
to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is 
, then in a move a player can't choose the number , but he can choose either or ). The player who first writes a perfect square on the board loses. Prove that one of the players has a winning strategy and determine who wins in the game.
such that 
for all 
such that 
.
identical cars on a circular track. Among all of them, they have just enough gas for one car to complete a lap. Show that there is a car that can complete a lap by collecting gas from the other cars on its way around
, which doesn't have right angle, let 
be its circumcenter. Let 
and 
be orthocenters of triangle 
and 
, respectively. Let 
be circumcenter of triangle 
. If circle 
and circle 
intersect at 
for the second time, prove that lies on 
.
is a polynomial with real coefficients such that![\[ P(x)^2 - 1 = 4 P(x^2 - 4x + 1). \]](http://latex.artofproblemsolving.com/b/4/2/b425a938a1dbbac25e0e0793b30c9e58845d45ab.png)
Find .
and 
be altitudes in an acute triangle 
which meet at 
. Suppose that 
meets the circumcircle of at 
and 
such that lies on the shorter arc of 
and lies on the shorter arc of 
. Let 
and 
meet at 
. Show that the circumcircles of 
and 
and the line 
concur.
be a diameter of a circle 
, and let 
be the tangent at 
. Furthermore, let 
be a fixed, positive real, and consider all pairs of points 
and 
lying on , on opposite sides of , such that 
. The lines 
and 
intersect at points 
and 
, respectively. Show that all the lines 
pass through a common point.
such that
for all 
satisfying 
, your right-neighbor’s is 
, and the next person over is 
, then for every trio in the circle they see
with circumcenter . 
and 
intersect the altitude from 
to at points and 
respectively. 
is the circumcenter of triangle 
and 
is the reflection of over . 
is the second intersection of circumcircles of triangles 
and 
. Show that 
are collinear.
at vertices 
and 
intersect at point 
and let 
be the point of intersection of 
and .Prove that 
if and only if 
.
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
and 
.
,
,
.
.
,
,
.
.
is cyclic, so 
.
.
:
.
.
,
.
and 
we have 
and 
,
and 
so 
.
:
.
,
,
,
,
.
and 
we have
Adding yields 
.
and 
,
.
so 
.