Prove that
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
high-quality questions. This test was created by multiple 130+ scorers, and we have tried our very best to produce the highest quality test possible. We've blended the question types in a good distribution, and maintained a realistic difficulty. (something most AOPS mocks fail at)
be the set of positive real numbers. Find all functions 
such that, for all 
,
Y by RedFireTruck
A polytope 
in the plane is called a lattice polytope if all its vertices lie on integral lattice points: those of the form 
with 
integers. It is known that there exist rational numbers 
and 
such that for any lattice polytope its area is area 
where
is the number of lattice points lying on the edges of and 
is the number of lattice points lying inside . [Thus in the example 
].
(i) Find and .
(ii) Prove the formula is valid for any lattice triangle.
in the plane is called a lattice polytope if all its vertices lie on integral lattice points: those of the form 
with 
integers. It is known that there exist rational numbers 
and 
such that for any lattice polytope its area is area 
where
is the number of lattice points lying on the edges of and 
is the number of lattice points lying inside . [Thus in the example 
].(i) Find
and .(ii) Prove the formula is valid for any lattice triangle.
This post has been edited 2 times. Last edited by parmenides51, Jan 13, 2021, 1:02 PM
Although I never really proved it, which I might try and post later if I can do it.
and the rectangle be 
.
for boundary, because that is the common convention.
and 
and 
.
