[/quote]
,
,![\[
1^2 + 2^2 + 3^2 + \ldots + k^2 = \frac{k(k + 1)(2k + 1)}{6}
\]](http://latex.artofproblemsolving.com/b/5/d/b5d49e25bd8d8a211b72372ff967558e1a8365ce.png)
Find the smallest positive integer 
is divisible by 200.
for real numbers 
not multiple of 
.
of the triangle 
is extended beyond 
to 
so that 
.
is extended beyond 
to 
.
,
.
Let 
.![$$ \frac{1}{2}\geq \frac{a}{a^2+3 }+ \frac{b}{b^2+3} \geq \frac{1}{4}(\frac{1}{\sqrt[3]{3}}+\sqrt[3]{3}-1)$$](http://latex.artofproblemsolving.com/f/9/1/f9119e071c12778037616c624c7b9ca75356e8e3.png)
![$$ \frac{1}{2}\geq \frac{a}{a^3+3 }+ \frac{b}{b^3+3} \geq \frac{1}{2\sqrt[3]{9}}$$](http://latex.artofproblemsolving.com/2/b/2/2b2d8a011229fe88a81cb75f9b018ad3a6245faa.png)
![$$ \frac{1}{2}\geq \frac{a}{a^2+ab+2}+ \frac{b}{b^2+ ab+2} \geq \frac{4\sqrt[3]{3}+3\sqrt[3]{9}-6}{17}$$](http://latex.artofproblemsolving.com/5/f/6/5f673cd494b1909ca237db63e96f949c24b6f2ad.png)
![$$ \frac{1}{2}\geq \frac{a}{a^3+ab+2}+ \frac{b}{b^3+ ab+2} \geq \frac{\sqrt[3]{3}}{5}$$](http://latex.artofproblemsolving.com/b/c/3/bc343a07c88dbd98ebffdfb78f311b7b82683d65.png)
and 
Prove that
Let 
Prove that
and 
are parallel. Angle bisectors of 
and 
meet at 
.
and 
meet at 
.
is cyclic
= 2∠AMP.
| 
with 
are positive integers and 
is prime 
)
Y by Adventure10, Mango247
Let 
be an interior point of the convex quadrilateral 
. Construct triangles 
and 
on the outside of the quadrilateral such that the similarities 
and 
hold. Let 
and 
be the projections of on the lines 
and 
, respectively. Prove that if the quadrilateral 
is cyclic, then
![\[EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.\]](//latex.artofproblemsolving.com/0/8/e/08e13ecbf07fb1e8950fa34e40cb7f08a8c5a7e1.png)
be an interior point of the convex quadrilateral 
. Construct triangles 
and 
on the outside of the quadrilateral such that the similarities 
and 
hold. Let 
and 
be the projections of on the lines 
and 
, respectively. Prove that if the quadrilateral 
is cyclic, then![\[EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.\]](http://latex.artofproblemsolving.com/0/8/e/08e13ecbf07fb1e8950fa34e40cb7f08a8c5a7e1.png)
and 
are similar.
= 
= 
and 
are similar, we get:
= 
= 
= 
(from similarities given)
= 
= 
.
= 
.
= 
+ 
= 
(from similarity of quadrilaterals 
+ 
= 
and 
,
,
,
and 
are cyclic.
we do the same for others, the claim follows.
,
and 
,
Similarly, 
,
and 
.
is clear since there is similar quadrilaterals. So let's prove that 
.
.
to 
.
to 
.
,
.
