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Part of 2015 VJIMC

Problems(2)

VJIMC 2015 Category I, Problem 4

Source: VJIMC2015

8/10/2015
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.
polynomialcollege contestsnumber theory
VJIMC 2015 Category II, Problem 4

Source: VJIMC2015

8/10/2015
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\}\ .
integrationcollege contestsreal analysis