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,![\[
1^2 + 2^2 + 3^2 + \ldots + k^2 = \frac{k(k + 1)(2k + 1)}{6}
\]](http://latex.artofproblemsolving.com/b/5/d/b5d49e25bd8d8a211b72372ff967558e1a8365ce.png)
Find the smallest positive integer such that 
is divisible by 200.
, Then total number of function from 
to such that 
be a trapezoid. By 
we denote the angle bisector of angle 
. Let 
and 
. Prove that 
is cyclic.
such that
Proposed by Ngo Van Trang, Vietnam
, since we can extend our given equation to 
. Next, note that the range of 
is ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
, so we want to be sure our value of gives a value for 
in the range . The only working value is 
, giving 
. We now require 
, and we can solve for to get 
.
has the properties that 
for all positive real values of 
for 
.
.
just consists of copies of the path 
"expanded" by 
every time 
is from the closest power of 
)
.
and 
the function reaches a maximum of 
is 
and 
we will have another 
.
.
