MathDB

3

Part of 2015 VJIMC

Problems(2)

VJIMC 2015 Category I, Problem 3

Source: VJIMC2015

8/10/2015
Problem 3 Let P(x)=x2015−2x2014+1 P(x) = x^{2015} -2x^{2014}+1 and Q(x)=x2015−2x2014−1 Q(x) = x^{2015} -2x^{2014}-1. Determine for each of the polynomials PP and QQ whether it is a divisor of some nonzero polynomial c0+c1x+…+cnxnc_0 + c_{1}x +\ldots + c_{n}x^n n whose coefficients cic_i are all in the set {−1,1} \{ -1, 1\}.
algebrapolynomialcollege contests
VJIMC 2015 Category II, Problem 3

Source: VJIMC2015

8/10/2015
Problem 3 Determine the set of real values of xx for which the following series converges, and find its sum: ∑n=1∞(∑k1,k2,…,kn≥01⋅k1+2⋅k2+…+n⋅kn=n(k1+…+kn)!k1!⋅…⋅kn!xk1+…+kn) .\sum_{n=1}^{\infty} \left(\sum_{\substack{k_1, k_2,\ldots , k_n \geq 0\\ 1\cdot k_1 + 2\cdot k_2+\ldots +n\cdot k_n = n}} \frac{(k_1+\ldots+k_n)!}{k_1!\cdot \ldots \cdot k_n!} x^{k_1+\ldots +k_n} \right) \ .
SummationConvergenceinfinite sumcollege contestsreal analysis