MathDB

P1

Part of 2022 Romania National Olympiad

Problems(4)

Romania NMO 2022 Grade 9 P1

Source: Romania National Olympiad 2022

4/21/2022
Let a,ba,b be positive integers. Prove that the equation x2+(a+b)2x+4ab=1x^2+(a+b)^2x+4ab=1 has rational solutions if and only if a=ba=b.
Mihai Opincariu
algebraromania
Romania NMO 2022 Grade 10 P1

Source: Romania National Olympiad 2022

4/21/2022
Let a≠1a\neq 1 be a positive real number. Find all real solutions to the equation ax=xx+log⁡a(log⁡a(x)).a^x=x^x+\log_a(\log_a(x)).
Mihai Opincariu
logarithmsalgebraromania
Romania NMO 2022 Grade 12 P1

Source: Romania National Olympiad 2022

4/20/2022
Let F\mathcal{F} be the set of functions f:R→Rf:\mathbb{R}\to\mathbb{R} such that f(2x)=f(x)f(2x)=f(x) for all x∈R.x\in\mathbb{R}.
[*]Determine all functions f∈Ff\in\mathcal{F} which admit antiderivatives on R.\mathbb{R}. [*]Give an example of a non-constant function f∈Ff\in\mathcal{F} which is integrable on any interval [a,b]⊂R[a,b]\subset\mathbb{R} and satisfies ∫abf(x) dx=0\int_a^bf(x) \ dx=0for all real numbers aa and b.b. Mihai Piticari and Sorin Rădulescu
functionIntegralromaniacalculus
Romania NMO 2022 Grade 11 P1

Source: Romania National Olympiad 2022

4/20/2022
Let f:[0,1]→(0,1)f:[0,1]\to(0,1) be a surjective function.
[*]Prove that ff has at least one point of discontinuity. [*]Given that ff admits a limit in any point of the interval [0,1],[0,1], show that is has at least two points of discontinuity. Mihai Piticari and Sorin Rădulescu
calculusromaniafunction