MathDB

M2661

Part of Kvant 2021

Problems(1)

Unordinary functional equation

Source: VI Caucasus Mathematical Olympiad

3/14/2021
An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions f1(x),f2(x),…f_1(x), f_2(x), \ldots let us place the number fi(j)f_i(j) into the cell (i,j)(i,j) of the table (for all i,j∈Ni, j\in \mathbb{N}). A sequence f1(x),f2(x),…f_1(x), f_2(x), \ldots is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions f1(x),f2(x),…f_1(x), f_2(x), \ldots , such that each fi(x)f_i(x) is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.
algebrafunctionsfunctional equation