5

Part of 2016 All-Russian Olympiad

Problems(2)

The last digits are zero...

Source: 2016 All-Russian Olympiad,Problem 9.5

Using each of the digits

1,2,3,\ldots ,8,9

exactly once,we form nine,not necassarily distinct,nine-digit numbers.Their sum ends in

n

n

is a non-negative integer.Determine the maximum possible value of

n

number theorySum

Polynomial with integer roots.

Source: All russian olympiad 2016,Day2,grade 11,P5

n

be a positive integer and let

k_0,k_1, \dots,k_{2n}

be nonzero integers such that

k_0+k_1 +\dots+k_{2n}\neq 0

. Is it always possible to a permutation

(a_0,a_1,\dots,a_{2n})

(k_0,k_1,\dots,k_{2n})

so that the equation \begin{align*} a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_0=0 \end{align*} has not integer roots?

number theoryInteger Polynomialalgebrapolynomial