MathDB

1

Part of 1989 Bundeswettbewerb Mathematik

Problems(2)

Bundeswettbewerb Mathematik 1989 Problem 1.1

Source: Bundeswettbewerb Mathematik 1989 Round 1

9/25/2022
For a given positive integer nn, let f(x)=xnf(x) =x^{n}. Is it possible for the decimal number 0.f(1)f(2)f(3)…0.f(1)f(2)f(3)\ldots to be rational? (Example: for n=2n=2, we are considering 0.1491625…0.1491625\ldots)
rationaldecimal representationperiodicnumber theory
Bundeswettbewerb Mathematik 1989 Problem 2.1

Source: Bundeswettbewerb Mathematik 1989 Round 2

9/23/2022
Determine the polynomial f(x)=xk+ak−1xk−1+⋯+a1x+a0f(x) = x^k + a_{k-1} x^{k-1}+\cdots +a_1 x +a_0 of smallest degree such that ai∈{−1,0,1}a_i \in \{-1,0,1\} for 0≤i≤k−10\leq i \leq k-1 and f(n)f(n) is divisible by 3030 for all positive integers nn.
algebrapolynomialnumber theoryDivisibilitydegree