be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer 
such that 
.
such that 
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
sastifying 
Y by CantonMathGuy, crazyeyemoody907, LoloChen
A triangle and a circle are in the same plane. Show that the area of the intersection of the triangle and the circle is at most one third of the area of the triangle plus one half of the area of the circle.
Krit Boonsiriseth
Krit Boonsiriseth
denote the triangle, 
denote the circle, and 
denote the reflection of the triangle in the center of the circle. Letting overline denote complement,![\begin{align*}
[\triangle \cap \circ] & \le \frac{1}{3}[\triangle] + \frac{1}{2}[\circ]\\
\iff [\triangle \cap \triangle' \cap \circ] + [\triangle \cap \overline{\triangle'} \cap \circ] & \le \frac{1}{3}[\triangle] + \frac{1}{2}\big[[\circ \cap \triangle \cap \triangle'] + 2[\circ \cap \triangle \cap \overline{\triangle'}] + [\circ \cap \overline{\triangle} \cap \overline{\triangle'}]\big]\\
\iff \frac{1}{2}[\triangle \cap \triangle' \cap \circ] & \le \frac{1}{3}[\triangle] + \frac{1}{2}[\circ \cap \overline{\triangle} \cap \overline{\triangle'}]\\
\iff [\triangle \cap \triangle' \cap \circ] & \le \frac{2}{3}[\triangle] + [\circ \cap \overline{\triangle} \cap \overline{\triangle'}].
\end{align*}](http://latex.artofproblemsolving.com/e/d/f/edf9a28ece344a7f6bb57b0a5a6c2035b63a1236.png)
Since 
is centrally symmetric, it's enough to show the following well-known lemma.
of a triangle can have area at most 
that of the triangle.
with medial triangle 
.
.
and two other vertices on 
and 
.
.
ratio with 
,
.
.
on the hexagon; by equal lengths, the similarity ratios 
,
,
sum to 
.
.
in 
are in the triangle. If neither 
,
and any ![$\lambda \in [0, 1]$](http://latex.artofproblemsolving.com/a/f/5/af534c9a7c6d2f6e7b850ef94556734f449b6297.png)
,
by
where 
.
is non-negative everywhere.![$([-20, 20]^2)^3$](http://latex.artofproblemsolving.com/e/2/2/e22292543c932aad4dc98d3bbae4b39870ef2c5b.png)
,
.
.
is greater than 
.![$[-20, 20]^2$](http://latex.artofproblemsolving.com/5/0/e/50e3ca820ae8eedc66494caed4efb376d5aadc61.png)
.
is non-empty, there is a point 
in 
.
be the height of the altitude onto this side. Note that 
.
,
as claimed. 
for which ![$R = ([-30, 30]^2)^3$](http://latex.artofproblemsolving.com/1/5/c/15c307e666d4041af63927a0d3ea9c281ea7e035.png)
.
is compact, and since 
,
.
,
of 
directly towards 
and the area of 
.![$[a, b] \subset [0, 1]$](http://latex.artofproblemsolving.com/0/6/3/063cf516e8b021cd123a7d1fd3f2300d25a90f39.png)
is the largest range such that 
is contained in ![$\lambda \in [a, b]$](http://latex.artofproblemsolving.com/2/3/a/23abbb9961842fec05c9f2af331da40dc30435bb.png)
.
Therefore in order to have a local minimum we must have 
.
being:
and hence we must have 
.
and 
,
.
and 
be the centre of 
be the distance between 
be half the length of 
,
.
.
