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 the point on 
such that 
. Let 
be the circle through 
tangent to 
at , and 
the circle through tangent to 
at 
. Let 
be the second intersection of and . Prove that 
lies on 
.
+1 w
is given. It is known that triangles 
, 
, 
, and 
are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle , lie on the same sphere.
and 
with integer coefficients is called 
if from the divisibility of both differences 
and 
by 
, it follows that both 
and 
are divisible by 100. Does there exist such an important pair of polynomials 
, 
, such that the pair 
and 
is also important?
+1 w
be real numbers such that 
. Show that
grid, taking turns. Petya goes first. On his turn, a player writes an uppercase Russian letter in an empty cell (each cell can contain only one letter). When all cells are filled, Petya is declared the winner if there are four consecutive cells horizontally spelling the word ``ПЕТЯ'' (PETYA) from left to right, or four consecutive cells vertically spelling ``ПЕТЯ'' from top to bottom. Can Petya guarantee a win regardless of Vasya's moves?
satisfy 
. Prove that
Also, find the values of for which the equality happens.
be an integer. What is the largest possible number of interior angles greater than 
in an 
-gon in the plane, given that the -gon does not intersect itself and all its sides have the same length?
Prove that
be reals such that 
and 
Prove that
satisfy:
![$[(x-2)^2+(y-4)^2][ (x-1)^2 + (y-2)^2] = 36$](http://latex.artofproblemsolving.com/b/1/0/b108dc2cac5b7491fb83418eb36c7184f6d5d30c.png)
such that: 
.
and 
.
and 
.
the answer is clearly 
.
a quadrilateral with four equal sides is a rhombus which is convex, so the answer is 
.
,
of these concave angles.
,
.
segments so that they become a bit longer and form an arc inside the original triangle. Then we also make the short side of the triangle slightly larger.
concave angles.
with 
we do the same with a triangle of sides 
(which exists since 
)
.