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.
board, show that there is a 
block on the board in which at least 3 of the squares are colored red.
+1 w
be a quadrilateral such that 
and 
are acute and 
. Suppose that 
, prove that 
.
be a square inscribed in a circle 
and let 
be a point on the shorter arc 
of . Let 
and 
and 
have equal areas.
towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns.
questions.
be any point on side 
of 
.Let 
and 
be points on and 
such that 
and 
respectively. Prove that the locus of circumcenter of 
is a line.
and 
pebbles, respectively. In the first round Barry chooses a positive integer 
and Andrew takes away 
pebbles from one of the two heaps (if is bigger than the number of pebbles in the heap, he takes away the complete heap). In the second round, the roles are reversed: Andrew chooses a positive integer and Barry takes away the pebbles from one of the two heaps. This goes on, in each round the two players are reversing the roles. The player that takes the last pebble loses the game.
be a triangle with 
. The point 
lies inside the triangle in such a way that 
. Let 
be the midpoint of . Prove that 
.
, is the midpoint of side , is the midpoint of side 
, is the midpoint of side .The line(different from line ) that is tangent to the inscribed circle of triangle and passing through point intersect line 
at 
.Define 
similarly.Prove that 
are collinear.
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let be the average of all the numbers in the grid. Prove that if 
the there exist two identical rows (meaning same numbers in the same order).
such that p is a prime, 
and 
is a divisor of 
.
) and not MG=CMsin(A-) !
(of the circumcircle) for which 
.
. From the relation 
results 
. This sistem has a unique solution. I know that 
, i.e. 
. In conclusion, 
, i.e. the point belongs to the diameter (which passes through 
) of the circumcircle.
is cyclic so 
.
.
is equivalent to the locus condition.
satisfies this condition. For any point 
on ray 
inside the triangle, 
similar to 
and 
similar to 
, similarly 
so also satisfies the condition. The locus is therefore the part of ray lying in the interior of .
![$$\left(a^{3}+b^{3}+c^{3}-3abc\right)^{2}-\left[a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a)\right]^{2}\ge 2k\cdot(a-b)^{2}(b-c)^{2}(c-a)^{2}$$](http://latex.artofproblemsolving.com/d/2/4/d24b46e65a587fbd977467b6e4e31838a0eb57aa.png)
forall 
where 
are the feet of the perpendiculars from 
is cyclic thus by 
theorem we have :
thus we have
and 
we have :
is a one-to-one function.
)
and 
denote distances bitween 
a
be a altitude.
and 
denote distances bitween 
a