MathDB

P3

Part of 2022 Romania National Olympiad

Problems(4)

Romania NMO 2022 Grade 9 P3

Source: Romania National Olympiad 2022

4/21/2022
Determine all functions f:R→Rf:\mathbb{R}\to\mathbb{R} for which there exists a function g:R→Rg:\mathbb{R}\to\mathbb{R} such that f(x)+f(y)=⌊g(x+y)⌋f(x)+f(y)=\lfloor g(x+y)\rfloor for all real numbers xx and yy.
Emil Vasile
functionalgebraromania
Romania NMO 2022 Grade 10 P3

Source: Romania National Olympiad 2022

4/21/2022
Let Z⊂CZ\subset \mathbb{C} be a set of nn complex numbers, n⩾2.n\geqslant 2. Prove that for any positive integer mm satisfying m⩽n/2m\leqslant n/2 there exists a subset UU of ZZ with mm elements such that∣∑z∈Uz∣⩽∣∑z∈Z∖Uz∣.\Bigg|\sum_{z\in U}z\Bigg|\leqslant\Bigg|\sum_{z\in Z\setminus U}z\Bigg|.Vasile Pop
complex numbersalgebraromania
Romania NMO 2022 Grade 11 P3

Source: Romania National Olympiad 2022

4/20/2022
Determine all functions f:R→Rf:\mathbb{R}\to\mathbb{R} which are differentiable in 00 and satisfy the following inequality for all real numbers x,yx,y f(x+y)+f(xy)≥f(x)+f(y).f(x+y)+f(xy)\geq f(x)+f(y).Dan Ștefan Marinescu and Mihai Piticari
romaniacalculusfunctionFunctional inequality
Romania NMO 2022 Grade 12 P3

Source: Romania National Olympiad 2022

4/20/2022
Let f,g:R→Rf,g:\mathbb{R}\to\mathbb{R} be two nondecreasing functions.
[*]Show that for any a∈R,a\in\mathbb{R}, b∈[f(a−0),f(a+0)]b\in[f(a-0),f(a+0)] and x∈R,x\in\mathbb{R}, the following inequality holds ∫axf(t) dt≥b(x−a).\int_a^xf(t) \ dt\geq b(x-a). [*]Given that [f(a−0),f(a+0)]∩[g(a−0),g(a+0)]≠∅[f(a-0),f(a+0)]\cap[g(a-0),g(a+0)]\neq\emptyset for any a∈R,a\in\mathbb{R}, prove that for any real numbers a<ba<b∫abf(t) dt=∫abg(t) dt.\int_a^b f(t) \ dt=\int_a^b g(t) \ dt.
Note: h(a−0)h(a-0) and h(a+0)h(a+0) denote the limits to the left and to the right respectively of a function hh at point a∈R.a\in\mathbb{R}.
calculusfunctionromania