MathDB

P4

Part of 2023 District Olympiad

Problems(4)

Just a normal FE

Source: Romanian District Olympiad 2023 10.4

3/11/2023
Determine all functions f:R→Rf:\mathbb{R}\to\mathbb{R} such that any real numbers xx{} and yy{} satisfy f(xf(x)+f(y))=f(f(x2))+y.f(xf(x)+f(y))=f(f(x^2))+y.
functional equationalgebraFunctional equation in R
NT Function with divisibility

Source: Romanian District Olympiad 2023 9.4

3/11/2023
Determine all strictly increasing functions f:N0→N0f:\mathbb{N}_0\to\mathbb{N}_0 which satisfy f(x)⋅f(y)∣(1+2x)⋅f(y)+(1+2y)⋅f(x)f(x)\cdot f(y)\mid (1+2x)\cdot f(y)+(1+2y)\cdot f(x)for all non-negative integers xx{} and yy{}.
functional equationnumber theory
Nice linear algebra

Source: Romanian District Olympiad 2023 11.4

3/11/2023
Let AA{} and BB{} be 3×33\times 3{} matrices with complex entries, satisfying A2=B2=O3A^2=B^2=O_3. Prove that if AA{} and BB{} commute, then AB=O3AB=O_3. Is the converse true?
linear algebramatrixrank
Three functions and a binary operation

Source: Romanian District Olympiad 2023 12.4

3/11/2023
Consider the functions f,g,h:R⩾0→R⩾0f,g,h:\mathbb{R}_{\geqslant 0}\to\mathbb{R}_{\geqslant 0} and the binary operation ∗:R⩾0×R⩾0→R⩾0*:\mathbb{R}_{\geqslant 0}\times \mathbb{R}_{\geqslant 0}\to \mathbb{R}_{\geqslant 0} defined as x∗y=f(x)+g(y)+h(x)⋅∣x−y∣,x*y=f(x)+g(y)+h(x)\cdot|x-y|,for all x,y∈R⩾0x,y\in\mathbb{R}_{\geqslant 0}. Suppose that (R⩾0,∗)(\mathbb{R}_{\geqslant 0},*) is a commutative monoid. Determine the functions f,g,hf,g,h.
real analysisabstract algebrafunction