MathDB

P2

Part of 2024 District Olympiad

Problems(4)

Sequences and fractional parts

Source: Romanian District Olympiad 2024 9.2

3/10/2024
Consider the sequence (an)n⩾1(a_n)_{n\geqslant 1} defined by a1=1/2a_1=1/2 and 2n⋅an+1=(n+1)an.2n\cdot a_{n+1}=(n+1)a_n. [*]Determine the general formula for an.a_n. [*]Let bn=a1+a2+⋯+an.b_n=a_1+a_2+\cdots+a_n. Prove that {bn}−{bn+1}≠{bn+1}−{bn+2}.\{b_n\}-\{b_{n+1}\}\neq \{b_{n+1}\}-\{b_{n+2}\}.
algebraSequencefractional part
weird sum for the orthocenters

Source: Romanian District Olympiad, grade 10, problem 2

3/10/2024
Let ABCABC be a triangle inscribed in the circle C(O,1)\mathcal{C}(O,1). Denote by s(M)=OH12+OH22+OH32,s(M)=OH_1^2+OH_2^2+OH_3^2, (∀)M∈C∖{A,B,C},(\forall) M \in\mathcal{C}\setminus \left\{A,B,C\right\}, where H1,H2,H3,H_1,H_2,H_3, are the orthocenters of the triangles MAB, MBCMAB,~MBC and MCA.MCA. a)a) Prove that if ABCABC is equilateral,, then s(M)=6,(∀)M∈C∖{A,B,C},s(M)=6,(\forall) M \in\mathcal{C}\setminus \left\{A,B,C\right\}, b)b) Prove that if there exist three distinct points M1,M2,M3∈C∖{A,B,C}M_1,M_2,M_3\in\mathcal{C}\setminus \left\{A,B,C\right\} such that s(M1)=s(M_1)=s(M2)s(M_2)=s(M3),=s(M_3), then ABCABC is equilateral..
geometrycomplex number geometry
Sequence with floor function

Source: Romanian District Olympiad 2024 11.2

3/10/2024
Let k⩾2k\geqslant 2 be an integer. Consider the sequence (xn)n⩾1(x_n)_{n\geqslant 1} defined by x1=a>0x_1=a>0 and xn+1=xn+⌊k/xn⌋x_{n+1}=x_n+\lfloor k/x_n\rfloor for n⩾1.n\geqslant 1. Prove that the sequence is convergent and determine its limit.
Sequenceslimitanalysis
Various derivatives and integrals

Source: Romanian District Olympiad 2024 12.2

3/10/2024
Let f:[0,1]→(0,∞)f:[0,1]\to(0,\infty) be a continous function on [0,1][0,1] and let A=∫01f(t)dt.A=\int_0^1 f(t)\mathrm{d}t. [*]Consider the function F:[0,1]→[0,A]F:[0,1]\to[0,A] defined by F(x)=∫0xf(t)dt.F(x)=\int_0^xf(t)\mathrm{d}t.Prove that F(x)F(x) has an inverse function, which is differentiable. [*]Prove that there exists a unique function g:[0,1]→[0,1]g:[0,1]\to[0,1] for which∫0xf(t)dt=∫g(x)1f(t)dt\int_0^xf(t)\mathrm{d}t=\int_{g(x)}^1f(t)\mathrm{d}tis satisfied for every x∈[0,1].x\in [0,1]. [*]Prove that there exists c∈[0,1]c\in[0,1] for whichlim⁡x→cg(x)−cx−c=−1,\lim_{x\to c}\frac{g(x)-c}{x-c}=-1,whre gg is the function uniquely determined at b.
Integralderivativereal analysis