[/quote]
,
and 
.
Where 
satisfies 
.
,
.
and a constant 
,
and 
are also friendly polynomials. Prove that 
.
lying outside the circle 
,
and 
.
is chosen on the segment 
.
and 
be the midpoints of segments 
and 
respectively. The circumcircle of triangle 
intersects 
)
passes through the centroid of triangle 
.
be an acute-angled triangle with 
and let 
be the foot of the
-
.
and 
in line 
and 
respectively. A line through 
and 
respectively such that 
while 
and 
and 
are tangent to each other.
with circumcircle 
.
is the angle bisector of 
.
intersects 
.
angle bisector of 
if and only if 
.
be real numbers satisfying
.
,
past 
.
such that ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
and 
intersect 
and 
at 
intersect 
.
with orthocenter 
intersects 
at 
intersect 
.
intersects 
and 
intersect on the circumcircle of triangle 
.
.![$D \in [AB]$](http://latex.artofproblemsolving.com/0/7/9/07914568284f3822cf15730a6a0e868f0bd79996.png)
is a point. Perpendiculars from ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
and ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
intersects with ![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
intersects with 
and 
.![$$[E_1F_1]=\frac{3}{4}$$](http://latex.artofproblemsolving.com/3/e/f/3efdf53cb3ff4b2a7fa60691f5aac9b3eeecf00c.png)
and 
.
be the point on 
.
Y by Adventure10, megarnie, Mango247
A box whose shape is a parallelepiped can be completely filled with cubes of side 
If we put in it the maximum possible number of cubes, each of volume 
, with the sides parallel to those of the box, then exactly 
percent of the volume of the box is occupied. Determine the possible dimensions of the box.
If we put in it the maximum possible number of cubes, each of volume 
, with the sides parallel to those of the box, then exactly 
percent of the volume of the box is occupied. Determine the possible dimensions of the box.
![\[ x = \left\lfloor\frac{a}{\sqrt[3]{2}}\right\rfloor \ \ \ \
y = \left\lfloor\frac{b}{\sqrt[3]{2}}\right\rfloor \ \ \ \
z = \left\lfloor\frac{c}{\sqrt[3]{2}}\right\rfloor \ \ \ \]](http://latex.artofproblemsolving.com/0/a/d/0ad67bde4ee74c09e63db306324de702f21d2751.png)
is the maximal number of cubes with sides of length ![$ \sqrt[3]{2}$](http://latex.artofproblemsolving.com/5/f/3/5f31adac765e62322c233a3f9f0da7a5157aee33.png)
that
.
,![\[ \frac ax\cdot \frac by\cdot \frac cz=5\ \ \ \ \ \ \ \ (1)\]](http://latex.artofproblemsolving.com/9/6/0/960d40c904c4d01ec487308bed476d8e01fabc06.png)
and 
for 
.
and 
.![\[ \begin{tabular}{|c|c|c|}
\hline
a & x & a/x \\ \hline
1 & 0 & - \\ \hline
2 & 1 & 2 \\ \hline
3 & 2 & 3/2 \\ \hline
4 & 3 & 4/3 \\ \hline
5 & 3 & 5/3 \\ \hline
6 & 4 & 3/2 \\ \hline
7 & 5 & 7/5 \\ \hline
8 & 6 & 4/3 \\
\hline
\end{tabular}\]](http://latex.artofproblemsolving.com/b/a/4/ba4d41cec5c40b5e625794c8194b423ecf38cde8.png)
:
and 
.
.![$ \frac ax\ge \sqrt[3]{5}$](http://latex.artofproblemsolving.com/1/c/c/1cc7a26e9415f483cfed8aab4c7a64a939985711.png)
.![\[ x< a/\sqrt[3]2 < x+1,\]](http://latex.artofproblemsolving.com/2/2/2/22282bcfa2c315e02a88fab0d24555b83e83d433.png)
![\[ \sqrt[3]2< a/x < \sqrt[3]2(1+\frac 1x)\]](http://latex.artofproblemsolving.com/2/d/a/2dab5f5b847e7944412e1ad54855f9993daeaeea.png)
then 
and ![$ \frac ax<\sqrt[3]2(1+\frac 1x)\le \sqrt[3]2(\frac 43)<\sqrt[3]5$](http://latex.artofproblemsolving.com/9/6/c/96c16643e10510c44a1a8428da091c24eff8865c.png)
.
and 
.![$ a/x=3/2<\sqrt[3]5$](http://latex.artofproblemsolving.com/0/2/1/021973fca4530cac7cc9e5f647841612d0fde603.png)
and so necessarily 
.![\[ \frac by \cdot \frac cz =\frac 52 \ \ \ \ \ \ (2)\]](http://latex.artofproblemsolving.com/2/1/4/2144675a9ba5255ed07b89ca0c2aae0f9fb14e3a.png)
imply ![\[ y< b/\sqrt[3]2 < y+1,\ \
\textrm{and} \ \ z< c/\sqrt[3]2 < z+1.\ \ \ \ (3)\]](http://latex.artofproblemsolving.com/2/8/3/2833fc55bd61aa71f73cfcaff46a3ab0d620a8d3.png)
that![\[ \frac by \ge \sqrt{5/2}\ \ \ \ \ (4)\]](http://latex.artofproblemsolving.com/0/b/8/0b8c820d050436ebf792122ddb8cf401229082b1.png)
then 
and we would have ![$ c/z=5/4<\sqrt[3]2$](http://latex.artofproblemsolving.com/4/6/5/465b7d23bed449d6d8981a5371915488cfdf4815.png)
,
.
then 
and ![$ \frac by<\sqrt[3]2(1+\frac 1y)\le \sqrt[3]2(\frac 54)<\sqrt{5/2}$](http://latex.artofproblemsolving.com/3/c/c/3cc39ccf0d1acfcd6d40aba780140161e34c7cbb.png)
as 
.
.
we have 
and for 
we have
and so necessarily 
and 
)
and 
.
and ![$ \frac cz<\sqrt[3]2(1+\frac 1z)\le \sqrt[3]2(\frac 76)<\frac 32$](http://latex.artofproblemsolving.com/7/1/c/71c6369719964ab482b413d7676206c831beee3c.png)
.
there are the only two possible values
and 
which yield the known solutions.
