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.
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples of real numbers, satisfying the equations, are such that 
are integers.
be a triangle, 
its circumcircle and 
the center of . Let 
be a point on the segment 
. We denote by 
the second intersection point of the circumcircles of triangles 
and 
. Prove that the line 
and the tangent to at point 
intersect on the circumcircle of triangle 
.
, 
, be positive real numbers such that 
. Prove that:![\[\frac{1}{(2a+b+c)^2}+\frac{1}{(a+2b+c)^2}+\frac{1}{(a+b+2c)^2}\leq \frac{3}{16}.\]](http://latex.artofproblemsolving.com/d/1/8/d18152c5635ee2f2059a860bf2fa362fe525ebb3.png)
satisfying the following two conditions:
such that, for all indices 
, 
.
and for any index , the number![\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]](http://latex.artofproblemsolving.com/5/4/a/54ac17063104439114f3c7b07de8fd27450ea1eb.png)
is a multiple of . be a circle with center . Let 
and 
be two fixed points on the circle and let be a variable point on . We denote by 
the intersection point of lines 
and 
, assuming 
. Let 
be the circumcircle of triangle 
. Let 
be the second intersection point of with . The tangent to at intersects at 
. The line 
intersects at 
. The perpendicular bisector of segment 
intersects line 
at 
, and line 
intersects at . We denote by 
the second intersection point of the circumcircle of triangle 
with . Prove that, as point varies, points and remain fixed.
into two regions, two parallel planes divide it into three regions, and two intersecting planes divide space into four regions. Consider the six planes which the faces of the cube 
lie on, and the four planes that the tetrahedron 
lie on. How many regions do these ten planes split the space into?
on the board. Charlotte has two operations available: the GCD operation and the LCM operation. and written on the board, erasing them, and writing the integer 
. and written on the board, erasing them, and writing the integer 
.
is called a winning number if there exists a sequence of operations such that, at the end, the only integer left on the board is . Find all winning integers among 
and, for each of them, determine the minimum number of GCD operations Charlotte must use. denotes the greatest common divisor of and , while the number denotes the least common multiple of and .
be an integer. We consider a square grid of size 
divided into 
unit squares. The grid is called balanced if:
, 
or 
.
.
, the smallest integer 
such that any balanced grid always contains an 
square whose absolute sum of the 
cells is less than or equal to 
.
, is circumscribed around the balls. Then equals
. The center of the semi-circle is the midpoint of segment 
. We have 
and 
... The points 
and 
divide 
into three equal parts while the points 
and 
divide 
into three equal parts. The line segments 
and 
intersect at . The line segments 
and 
intersect at . Given that the areas of the triangles 
, 
and 
are 
, 
, and 
respectively, find the area of the triangle 
.
, where 
are real numbers. Suppose that 
has three distinct real roots and the polynomial 
has no real roots. Prove that![\[
P(1) > 69^3.
\]](http://latex.artofproblemsolving.com/2/a/4/2a49f9db6af78d720d5a657737625c011a1b554a.png)
such that 
.
has no real roots, and 
takes all real values 
,
.
,
,
--- and so