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Part of 2010 Bundeswettbewerb Mathematik

Problems(2)

2010 different ways as sum of powers of 2

Source: Germany Federal - Bundeswettbewerb Mathematik 2010, round 1, p4

4/10/2020
Find all numbers that can be expressed in exactly 20102010 different ways as the sum of powers of two with non-negative exponents, each power appearing as a summand at most three times. A sum can also be made from just one summand.
power of 2Sum of powerscombinatorics
f (f (f (n))) = P (n), for all n \in N_0, all coefficients of P(x) are in N_0

Source: Germany Federal - Bundeswettbewerb Mathematik 2010, round 2, p4

4/14/2020
In the following, let N0N_0 denotes the set of non-negative integers. Find all polynomials P(x)P(x) that fulfill the following two properties: (1) All coefficients of P(x)P(x) are from N0N_0. (2) Exists a function f:N0→N0f : N_0 \to N_0 such as f(f(f(n)))=P(n)f (f (f (n))) = P (n) for all n∈N0n \in N_0.
polynomialfunctionalgebrafunctional equationfunctional