or 
tile. Determine in how many ways exactly 
dominoes can be placed without overlapping on a 
chessboard so that every 
square contains at least two uncovered unit squares which lie in the same row or column.
.![\[
\boxed{
\frac{a^5}{b^2} + \frac{b}{c} + \frac{c^3}{a^2} > 2a
}
\]](http://latex.artofproblemsolving.com/7/a/a/7aacf59915d9c5477d038e045bf017f26ec38e3c.png)
![\[
\frac{a^5}{b^2} + \frac{b}{c} + \frac{c^3}{a^2}
=
3 \cdot \frac{a^5}{3b^2}
+ 6 \cdot \frac{b}{6c}
+ 2 \cdot \frac{c^3}{2a^2}.
\]](http://latex.artofproblemsolving.com/2/b/2/2b2975adad3be71344c3362638e4ee558c9c3252.png)
![\[
\frac{a^5}{b^2} + \frac{b}{c} + \frac{c^3}{a^2}
\ge
11 \cdot
\sqrt[11]{\left( \frac{a^5}{3b^2} \right)^3
\cdot \left( \frac{b}{6c} \right)^6
\cdot \left( \frac{c^3}{2a^2} \right)^2}.
\]](http://latex.artofproblemsolving.com/c/4/5/c45c9d0a538e9fa71feed82e329137dd2cd3012f.png)
![\[
\left( \frac{a^5}{3b^2} \right)^3
= \frac{a^{15}}{3^3 b^6}, \qquad
\left( \frac{b}{6c} \right)^6
= \frac{b^6}{6^6 c^6}, \qquad
\left( \frac{c^3}{2a^2} \right)^2
= \frac{c^6}{2^2 a^4}.
\]](http://latex.artofproblemsolving.com/9/2/f/92fb421b88f5c4c485128034fe128115a8fddb7b.png)
![\[
\frac{a^{15}}{3^3 b^6} \cdot
\frac{b^6}{6^6 c^6} \cdot
\frac{c^6}{2^2 a^4}
=
\frac{a^{15 - 4} \cdot b^6 \cdot c^6}
{3^3 \cdot 6^6 \cdot 2^2 \cdot b^6 \cdot c^6}
=
\frac{a^{11}}{3^3 \cdot 2^2 \cdot 6^6}.
\]](http://latex.artofproblemsolving.com/e/8/b/e8b8476462266a6862295db3859b305efbd2fff5.png)
![\[
3^3 \cdot 2^2 \cdot 6^6
= 27 \cdot 4 \cdot (2 \cdot 3)^6
= 27 \cdot 4 \cdot 2^6 \cdot 3^6
= 2^8 \cdot 3^9.
\]](http://latex.artofproblemsolving.com/6/3/e/63e0ce7d657232de16105f97a76123db324f2d25.png)
![\[
\frac{a^5}{b^2} + \frac{b}{c} + \frac{c^3}{a^2}
\ge
11 \cdot \sqrt[11]{\frac{a^{11}}{2^8 \cdot 3^9}}
= \frac{11a}{\sqrt[11]{2^8 \cdot 3^9}}.
\]](http://latex.artofproblemsolving.com/a/5/f/a5faa7928603e9fec0f76d206ab90460786b01f1.png)
![\[
\sqrt[11]{2^8 \cdot 3^9}
< \frac{11}{2} \quad \Rightarrow \quad
\frac{11a}{\sqrt[11]{2^8 \cdot 3^9}} > 2a.
\]](http://latex.artofproblemsolving.com/0/f/8/0f8c86b4e015becf7292c8871b12ec28ad590aa7.png)
![\[
\boxed{
\frac{a^5}{b^2} + \frac{b}{c} + \frac{c^3}{a^2} > 2a
}
\qquad \text{for all } a,b,c>0.
\]](http://latex.artofproblemsolving.com/4/2/8/428d46758a2648537f0b121fa8f923a510c030c7.png)
such that 
Y by Adventure10, Mango247
Two circles 
and 
intersect at 
and 
. The common tangent of and that is close to than meets at 
and at 
. 
extended meets at 
and 
extended meets at 
. Let 
and 
be the midpoints of 
and 
respectively.
Show that
bisects angle 
.
and 
intersect at 
and 
. The common tangent of and that is close to than meets at 
and at 
. 
extended meets at 
and 
extended meets at 
. Let 
and 
be the midpoints of 
and 
respectively.Show that
bisects angle 
.
,
,
and 
.
concyclic. Similarly, 
concyclic. Then 
.
.
and 
.
.
,
and 
are three similar triangles, and 
are their respective medians. The result follows.
