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.
an 
-division of 
if
,
.
such that for any 
-division 
of 
, there exists a set 
(
) such that there are two elements 
of such that 
.
be an integer and let 
be non-negative real numbers. Define 
for 
. Prove that![\[\frac{1}{n} \sum_{k=0}^{n-1} S_k^2-\frac{1}{n^2}\left(\sum_{k=0}^{n} S_k\right)^2\le \frac{4}{45} (S_n-S_0)^2.\]](http://latex.artofproblemsolving.com/4/6/9/4693cc8ec828972286e270c630ef06e7d57d3a8c.png)
be a triangle, and let 
be centered at 
, 
and tangent to line 
at 
, 
respectively. Let line 
intersect 
again at 
and let line 
intersect 
again at 
. If 
is the other intersection of the circumcircles of triangles and 
, then prove that lines 
, , and 
either concur or are all parallel.
, such that for 
![\[
\frac{(xy - 4)(x^2 + y^2)}{xy(x + y - 2\sqrt{2})^2}
\]](http://latex.artofproblemsolving.com/d/5/0/d503a5fb7f77298fb09d6e1696adeee12063305f.png)
for all 
, 
, .
, 
and 
. Find the area of 
if the area of is 
.
by trimming its four corners. Find the area of the octagon., triangles 
, and 
have areas of 
, 
, and 
, respectively. What is the area of triangle 
? (See figure in problem 1.) with 
, a point 
is taken on side such that 
. Let 
be the foot of the perpendicular lowered from to line 
. Find 
.
. Find the radius of the circle that passes through points 
, and .
is tangent to the four sides of a trapezoid where and 
are parallel. If 
cm and the area of is 
cm
, determine the length of 
. we have 
and 
. Let be a point on such that 
. Prove that 
. is divided into 
right triangles. Show that 
satisfies the equation![\[x^5 - x^4 + 3x^3 - 2x^2 + 3x - 1 = 0.\]](http://latex.artofproblemsolving.com/3/c/5/3c59003a67854924409508d1781319767aeac3a9.png)
, 
. Determine the maximum value of 
, such that![\[\sum_{t=1}^n\frac{\lfloor x_t\rfloor ^2}{x_t}\ge\lambda\]](http://latex.artofproblemsolving.com/5/3/a/53aacf7af816194fe9c73b488ecf8379a368db95.png)
holds for all 
with 
then prove that
When does equality hold?
and 
be different points on a circle 
such that 
is not a diameter. Let 
be the tangent line to at . Point 
is such that is the midpoint of the line segment 
. Point 
is chosen on the shorter arc of so that the circumcircle 
of triangle 
intersects at two distinct points. Let 
be the common point of and that is closer to . Line 
meets again at 
. Prove that the line 
is tangent to . is marked on side of triangle so that 
. It turned out that 
and 
. Find 
.
,