MathDB
Representation in ternary system

Source: AIMO 2007, TST 5, P1

January 11, 2009
algebrapolynomialnumber theory unsolvednumber theory

Problem Statement

Let k∈N k \in \mathbb{N}. A polynomial is called k k-valid if all its coefficients are integers between 0 and k k inclusively. (Here we don't consider 0 to be a natural number.) a.) For n∈N n \in \mathbb{N} let an a_n be the number of 5-valid polynomials p p which satisfy p(3)=n. p(3) = n. Prove that each natural number occurs in the sequence (an)n (a_n)_n at least once but only finitely often. b.) For n∈N n \in \mathbb{N} let an a_n be the number of 4-valid polynomials p p which satisfy p(3)=n. p(3) = n. Prove that each natural number occurs infinitely often in the sequence (an)n (a_n)_n .
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