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 
for which there exists a natural number
such that the number 
is integer for all natural 
.
has 
as the midpoint of 
and orthocenter 
. Let 
be an arbitrary point on the nine point circle of 
. The line through perpendicular to 
intersects at 
. The line through 
perpendicular to 
intersects 
at 
. If 
is the midpoint of , show that 
.
that, for all and positive real numbers 
, there is![\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]](http://latex.artofproblemsolving.com/4/8/b/48b9fc4af5a20de7fd9dfc45a74d5aa27e615e46.png)
that satisfy the condition![\[
\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 3.
\]](http://latex.artofproblemsolving.com/2/7/d/27d266570e3a105a4b416bb7c55904ffabf7a07c.png)
Find the maximum value of the expression![\[
P = \dfrac{yz}{\sqrt[3]{3y^2z^2+ 3x^2y^2z^2+ x^2z^2 + x^2y^2}}
+ \frac{xz}{\sqrt[3]{3x^2z^2 + 3x^2y^2z^2 + x^2y^2 + y^2z^2}}
+ \frac{xy}{\sqrt[3]{3x^2y^2 + 3x^2y^2z^2 +y^2z^2 + x^2z^2}}.
\]](http://latex.artofproblemsolving.com/b/a/f/bafc95e59d2f28dbb4e02771db38ce736e3fd45c.png)
-sequence is a sequence of 
numbers 
each equal to either 
or 
. Determine the largest 
so that, for any -sequence, there exists an integer 
and indices 
so that 
for all 
, and 
and 
such that
be reals such that 
and 
Find the value of 
players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least 
next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.
and 
Find the value of 
be a triangle and , 
, 
be the midpoints of arcs , 
, 
on the circumcircle. Line 
passes through the feet of the perpendiculars from to 
and 
. Line 
passes through the feet of the perpendiculars from to and 
. Let 
denote the intersection of lines and . Define points 
and 
similarly. Prove that triangle 
and 
are similar to each other.
is a sphere with a diameter AP and passing through the point F. This sphere is centered on the midline M, N of the triangle parallel to the side BC, where M, N are the midpoints of the sides AB, AC. If the point A lies on the line BC, then the points A, F and the locus of the vertices V of the right angles is a sphere centered on segment M, N, where M, N are the midpoints of the segments AB, AC. As the point P moves on the segment BC, the locus of the vertices V becomes a portion of the pencil of spheres centered on the segment MN and passing through the points A, F. If the point A does not lie on the line BC, the pencil of spheres is elliptic (the spheres intersect in a fixed circle with the diameter AF and perpendicular to the line BC). If the point A lies on the line BC, the pencil of spheres is parabolic (the spheres touch the plane perpendicular to the line BC at the point A).
be 
Therefore, given and 
all possible vertices of the right angle lie on the circle with diameter 
Thus, when all possible circles are drawn, it encloses a region as shown in the diagram below:![[asy]
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draw(circle((6.200000,3.000000),3.104835));
draw(circle((6.250000,3.000000),3.092329));
draw(circle((6.300000,3.000000),3.080584));
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draw(circle((6.400000,3.000000),3.059412));
draw(circle((6.450000,3.000000),3.050000));
draw(circle((6.500000,3.000000),3.041381));
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draw(circle((6.750000,3.000000),3.010399));
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draw(circle((6.900000,3.000000),3.001666));
draw(circle((6.950000,3.000000),3.000417));
draw(circle((7.000000,3.000000),3.000000));
draw(circle((7.050000,3.000000),3.000417));
draw(circle((7.100000,3.000000),3.001666));
draw(circle((7.150000,3.000000),3.003748));
draw(circle((7.200000,3.000000),3.006659));
draw(circle((7.250000,3.000000),3.010399));
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draw(circle((7.700000,3.000000),3.080584));
draw(circle((7.750000,3.000000),3.092329));
draw(circle((7.800000,3.000000),3.104835));
draw(circle((7.850000,3.000000),3.118092));
draw(circle((7.900000,3.000000),3.132092));
draw(circle((7.950000,3.000000),3.146824));
draw(circle((8.000000,3.000000),3.162278));
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draw(circle((8.150000,3.000000),3.212865));
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draw(circle((8.300000,3.000000),3.269557));
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draw(circle((8.400000,3.000000),3.310589));
draw(circle((8.450000,3.000000),3.332041));
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dot((0,0),red);
dot((10,0),red);
dot((7,6),red);
label((0,0),"$B$",SW,red);
label((10,0),"$C$",SE,red);
label((7,6),"$A$",N,red);
draw((0,0)--(7,6)--(10,0)--cycle,red);
[/asy]](http://latex.artofproblemsolving.com/c/a/2/ca2b9e2b05e066faf4f85049352b24def792b887.png)
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