[/quote]
,
of the pizza, where 
is a positive integer. Find 
![\[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]](http://latex.artofproblemsolving.com/e/b/2/eb28918359848dd07784a3c2c6a34b934743c016.png)
be 
.![\[\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}\]](http://latex.artofproblemsolving.com/3/1/5/315341a7adf35bc04e40ee0685a011c9f7b76115.png)
without Newton Sums.
and 
Prove that
Is there an elegant way to prove this?
and incentre 
.
intersects the lines 
,
,
at points 
,
,
,
,
,
,
,
,
,
.
,
.
![$\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
denote the set of single-variable polynomials in ![$\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/0/3/2/032a637d4ae21da2c0ddd8dc127723807f11cc5f.png)
(i.e. functions taking polynomials to polynomials)![$p, q \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/7/e/7/7e7eb925258a33e456ef4ff722c1f943dc155d10.png)
,
;![$p \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/0/2/6/0267b98c16b400dda1f081877e9b84fe672ce0e7.png)
,
has an integer root if and only if 
does.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
that verify the following equation for all 
:
is a function for which the expression 
for all 
is always a perfect square. Prove that 
for all 
.
such that 
.
be a quadrilateral inscribed in a circle 
,
be a point on the extension of 
such that 
and 
are tangent to 
and the line 
.
and 
and 
at 
and 
the reflections of 
with 
,
be the midpoint of 
and 
are perpendicular.
.
be the feet of the altitudes from 
to 
respectively. Let 
be the circle with diameter 
be the intersection of 
and 
be the intersection of 
be the lines through 
respectively. Let 
and 
.
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
Y by
(a) The Reuleaux triangle, constructed from an equilateral triangle, is the union of three arcs of a circle of amplitude 
(see the figure on the left). Show that its width is the same in all directions (see the figure in the center).
(b) We want to form other curves of constant width, from a triangle
whose sides measure 
, and 
by drawing six arcs of a circle as in the figure on the right: each side of the triangle is extended in both directions; each vertex becomes the center of two arcs of a circle "on either side" whose radii, for example around 
, are equal to 
and 
.
i) For these values of
and 
, show that there exists at least one choice of values of 
and 
which gives a continuous curve.
ii) For these values of and what are all the value choices for and which give a continuous curve?
iii) What is the perimeter of the continuous figure thus drawn? Show that its width is the same in all directions.
(see the figure on the left). Show that its width is the same in all directions (see the figure in the center).
(b) We want to form other curves of constant width, from a triangle
whose sides measure 
, and 
by drawing six arcs of a circle as in the figure on the right: each side of the triangle is extended in both directions; each vertex becomes the center of two arcs of a circle "on either side" whose radii, for example around 
, are equal to 
and 
.i) For these values of
and 
, show that there exists at least one choice of values of 
and 
which gives a continuous curve.ii) For these values of
and what are all the value choices for and which give a continuous curve?iii) What is the perimeter of the continuous figure thus drawn? Show that its width is the same in all directions.
This post has been edited 2 times. Last edited by parmenides51, Dec 20, 2020, 3:40 AM
and a point 
with 
the lenght of the side of the equilateral triangle.
and the distance 
.
.
with 
a real positive number.
.
.
