MathDB

Let

n

be a given positive integer.
(a) Do there exist

2n+1

consecutive positive integers

a_0,a_1,\ldots,a_{2n}

in the ascending order such that

a_0+a_1+\ldots+a_n=a_{n+1}+\ldots+a_{2n}

? (b) Do there exist consecutive positive integers

a_0,a+1,\ldots,a_{2n}

in ascending order such that

a_0^2+a_1^2+\ldots+a_n^2=a_{n+1}^2+\ldots+a_{2n}^2

? (c) Do there exist consecutive positive integers

a_0,a_1,\ldots,a_{2n}

in ascending order such that

a_0^3+a_1^3+\ldots+a_n^3=a_{n+1}^3+\ldots+a_{2n}^3

?
[hide=Official Hint]You may study the function

f(x)=(x-n)^3+\ldots+x^3-(x+1)^3-\ldots-(x+n)^3

and prove that the equation

f(x)=0

has a unique solution

x_n

with

3n(n+1)<x_n<3n(n+1)+1

. You may use the identity

1^3+2^3+\ldots+n^3=\frac{n^2(n+1)^2}2

Problem 2