MathDB

2015 VJIMC

Part of Vojtěch Jarník IMC

Subcontests

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VJIMC 2015 Category I, Problem 4

Problem 4 Let mm be a positive integer and let pp be a prime divisor of mm. Suppose that the complex polynomial a0+a1x+…+anxna_0 + a_1x + \ldots + a_nx^n with n<pp−1φ(m)n < \frac{p}{p-1}\varphi(m) and an≠0a_n \neq 0 is divisible by the cyclotomic polynomial ϕm(x)\phi_m(x). Prove that there are at least pp nonzero coefficients ai .a_i\ .
The cyclotomic polynomial ϕm(x)\phi_m(x) is the monic polynomial whose roots are the mm-th primitive complex roots of unity. Euler’s totient function φ(m)\varphi(m) denotes the number of positive integers less than or equal to mm which are coprime to mm.

VJIMC 2015 Category II, Problem 4

Problem 4 Find all continuously differentiable functions f:R→R f : \mathbb{R} \rightarrow \mathbb{R} , such that for every a≥0a \geq 0 the following relation holds: ∭D(a)xf(ayx2+y2) dx dy dz=πa38(f(a)+sin⁡a−1) ,\iiint \limits_{D(a)} xf \left( \frac{ay}{\sqrt{x^2+y^2}} \right) \ dx \ dy\ dz = \frac{\pi a^3}{8} (f(a) + \sin a -1)\ , where D(a)={(x,y,z) : x2+y2+z2≤a2 , ∣y∣≤x3} .D(a) = \left\{ (x,y,z)\ :\ x^2+y^2+z^2 \leq a^2\ , \ |y|\leq \frac{x}{\sqrt{3}} \right\}\ .
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VJIMC 2015 Category I, Problem 3

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\}.

VJIMC 2015 Category II, Problem 3

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) \ .
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