MathDB

3

Part of 2019 Centers of Excellency of Suceava

Problems(3)

Geometric equality

Source:

7/7/2020
The circumcenter, circumradius and orthocenter of a triangle ABC ABC satisfying AB<AC AB<AC are notated with O,R,H, O,R,H, respectively. Prove that the middle of the segment OH OH belongs to the line BC BC if AC2−AB2=2R⋅BC. AC^2-AB^2=2R\cdot BC.
Marius Marchitan
rotationgeometryorthocentercircumcircle
On the relation between some arithmetic and geometric progressions

Source:

7/7/2020
Let (an)n≥1 \left( a_n \right)_{n\ge 1} be a non-constant arithmetic progression of positive numbers and (gn)n≥1 \left( g_n \right)_{n\ge 1} be a non-constant geometric progression of positive numbers satisfying a1=g1 a_1=g_1 and a2019=g2019. a_{2019} =g_{2019} . Specify the set {k∈N∣ak≤gk} \left\{ k\in\mathbb{N} \big| a_k\le g_k \right\} and prove that it bijects the natural numbers.
Gheorghe Rotariu
arithmetic sequencegeometric sequencebijectionpermutation of the naturalsalgebra
On functions f,g satisfying (fg)'=f'g'

Source:

7/7/2020
For two real intervals I,J, I,J, we say that two functions f,g:I⟶J f,g:I\longrightarrow J have property P \mathcal{P} if they are differentiable and (fg)′=f′g′. (fg)'=f'g'.
a) Provide example of two nonconstant functions a,b:R⟶R a,b:\mathbb{R}\longrightarrow\mathbb{R} that have property P. \mathcal{P} .
b) Find the functions λ:(2019,∞)⟶(0,∞) \lambda :(2019,\infty )\longrightarrow (0,\infty ) having the property that λ \lambda along with θ:(2019,∞)⟶(0,∞),θ(x)=x2019 \theta :(2019,\infty )\longrightarrow (0,\infty ), \theta (x)=x^{2019} have property P. \mathcal{P} .
Dan Nedeianu
functionreal analysisdifferentiability