MathDB

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Part of 2019 VJIMC

Problems(2)

VJIMC 2019 P2

Source: VJIMC 2019

3/29/2019
A triplet of polynomials u,v,w∈R[x,y,z]u,v,w \in \mathbb{R}[x,y,z] is called smart if there exists polynomials P,Q,R∈R[x,y,z]P,Q,R\in \mathbb{R}[x,y,z] such that the following polynomial identity holds :u2019P+v2019Q+w2019R=2019u^{2019}P +v^{2019 }Q+w^{2019} R=2019
a) Is the triplet of polynomials u=x+2y+3,        v=y+z+2,          w=x+y+zu=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y+z smart? b) Is the triplet of polynomials u=x+2y+3,        v=y+z+2,          w=x+y−zu=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y-z smart?
Proposed by Arturas Dubickas (Vilnius University).
algebrapolynomialreal analysisVJIMC2019VJIMCVojtech JarnikAnnual Vojtech Jarnic
VJIMC 2019 P2 Category II

Source: VJIMC 2019

3/30/2019
Find all twice differentiable functions f:R→Rf : \mathbb{R} \to \mathbb{R} such that f′′(x)cos⁡(f(x))≥(f′(x))2sin⁡(f(x))f''(x) \cos(f(x))\geq(f'(x))^2 \sin(f(x)) for every x∈Rx\in \mathbb{R}.
Proposed by Orif Ibrogimov (Czech Technical University of Prague), Karim Rakhimov (University of Pisa)
real analysisfunctional equationVJIMC2019VJIMCVojtech JarnikAnnual Vojtech Jarnic