MathDB

3

Part of 2010 Laurențiu Panaitopol, Tulcea

Problems(4)

An angle invariant to a certain movable circle

Source:

10/28/2019
Let R R be the circumradius of a triangle ABC. ABC. The points B,C, B,C, lie on a circle of radius ρ \rho that intersects AB,AC AB,AC at E,D, E,D, respectively. ρ′ \rho' is the circumradius of ADE. ADE. Show that there exists a triangle with sides R,ρ,ρ′, R,\rho ,\rho' , and having an angle whose value doesn't depend on ρ. \rho .
Laurențiu Panaitopol
geometrycircumcircleinvariantLocus
Re(z^n)>Im(z^n), for any n, implies z is a positive real

Source:

10/28/2019
Let be a complex number z z having the property that ℜ(zn)>ℑ(zn), \Re \left( z^n \right) >\Im \left( z^n \right) , for any natural numbers n. n. Show that z z is a positive real number.
Laurențiu Panaitopol
complex numbersalgebra
A sufficient condition for a function to be of class C^infinity

Source:

10/28/2019
Let be a twice-differentiable function f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} that has the properties that: (i) suppf′′=f(R) \text{(i) supp} f''=f\left(\mathbb{R}\right) \text{(ii)}\exists g:\mathbb{R}\longrightarrow\mathbb{R} \forall x\in\mathbb{R}  f(x+1)=f(x)+f'\left( g(x)\right)\text{ and } f'(x+1)=f'(x)+f''\left( g(x)\right)
Prove that: a) any such g g is injective. b) f f is of class C∞, C^{\infty } , and for any natural number n, n, any real number x x and any such g, g, f(n)(x+1)=f(n)(x)+f(n+1)(g(x)).f^{(n)}(x+1)=f^{(n)}(x)+f^{(n+1)}\left( g(x)\right) .
Laurențiu Panaitopol
functionreal analysiscalculusderivativeSupportmonotony
A nice problem with polynoms by Panaitopol

Source:

10/28/2019
Let be two polynoms P,Q∈R[X] P,Q\in\mathbb{R} [X] having the property that ∣{n∈{0}∪N∣P(n)≤Q(n)}∣=∣{n∈{0}∪N∣P(n)≥Q(n)}∣=∞. \left| \{ n\in\{ 0\}\cup\mathbb{N} | P(n)\le Q(n) \} \right| =\left| \{ n\in\{ 0\}\cup\mathbb{N} | P(n)\ge Q(n) \} \right| =\infty . Show that P=Q. P=Q.
Laurențiu Panaitopol
polynomsalgebra