MathDB

P1

Part of 2022 District Olympiad

Problems(4)

(x^3+3x^2f(y))/(x+f(y))+(y^3+3y^2f(x))/(y+f(x))=(x+y)^3/f(x+y)

Source: Romanian District Olympiad 2022 - Grade 9 - Problem 1

3/27/2022
Let f:N∗→N∗f:\mathbb{N}^*\rightarrow \mathbb{N}^* be a function such that x3+3x2f(y)x+f(y)+y3+3y2f(x)y+f(x)=(x+y)3f(x+y), (∀)x,y∈N∗.\frac{x^3+3x^2f(y)}{x+f(y)}+\frac{y^3+3y^2f(x)}{y+f(x)}=\frac{(x+y)^3}{f(x+y)},~(\forall)x,y\in\mathbb{N}^*. a)a) Prove that f(1)=1.f(1)=1. b)b) Find function f.f.
Arithmetic Functionfunctional equationfunctionalgebra
Romania District MO 2022 Grade 10 P1

Source: Romania District MO 2022 Grade 10

3/28/2022
Determine all x∈(0,3/4)x\in(0,3/4) which satisfy log⁡x(1−x)+log⁡21−xx=1(log⁡2x)2.\log_x(1-x)+\log_2\frac{1-x}{x}=\frac{1}{(\log_2x)^2}.
romanialogarithmsalgebra
Romania District MO 2022 Grade 11 P1

Source: Romania District MO 2022 Grade 11

3/27/2022
Let f,g:R→Rf,g:\mathbb{R}\to\mathbb{R} be functions which satisfy inf⁡x>af(x)=g(a) and sup⁡x<ag(x)=f(a),\inf_{x>a}f(x)=g(a)\text{ and }\sup_{x<a}g(x)=f(a),for all a∈R.a\in\mathbb{R}. Given that ff has Darboux's Property (intermediate value property), show that functions ff and gg are continuous and equal to each other.
Mathematical Gazette
darboux s propertyfunctioncollege contestsromania
Romania District MO 2022 Grade 12 P1

Source: Romania District MO 2022 Grade 12

3/27/2022
Let ee be the identity of monoid (M,⋅)(M,\cdot) and a∈Ma\in M an invertible element. Prove that
[*]The set Ma:={x∈M:ax2a=e}M_a:=\{x\in M:ax^2a=e\} is nonempty; [*]If b∈Mab\in M_a is invertible, then b−1∈Mab^{-1}\in M_a if and only if a4=ea^4=e; [*]If (Ma,⋅)(M_a,\cdot) is a monoid, then x2=ex^2=e for all x∈Ma.x\in M_a.
Mathematical Gazette
romaniamonoidcollege contests