MathDB

P3

Part of 2022 District Olympiad

Problems(4)

On x+3^y and y+3^x.

Source: Romanian District Olympiad 2022 - Grade 9 - Problem 3

3/27/2022
a)a) Solve over the positive integers 3x=x+2.3^x=x+2. b)b) Find pairs (x,y)∈N×N(x,y)\in\mathbb{N}\times\mathbb{N} such that (x+3y)(x+3^y) and (y+3x)(y+3^x) are consecutive.
number theoryDiophantine equationinduction
Romania District MO 2022 Grade 11 P3

Source: Romania District MO 2022 Grade 11

3/27/2022
Let (xn)n≥1(x_n)_{n\geq 1} be the sequence defined recursively as such: x1=1, xn+1=x1n+1+x2n+2+⋯+xn2n ∀n≥1.x_1=1, \ x_{n+1}=\frac{x_1}{n+1}+\frac{x_2}{n+2}+\cdots+\frac{x_n}{2n} \ \forall n\geq 1.Consider the sequence (yn)n≥1(y_n)_{n\geq 1} such that yn=(x12+x22+⋯xn2)/ny_n=(x_1^2+x_2^2+\cdots x_n^2)/n for all n≥1.n\geq 1. Prove that
[*]xn+12<yn/2x_{n+1}^2<y_n/2 and yn+1<(2n+1)/(2n+2)⋅yny_{n+1}<(2n+1)/(2n+2)\cdot y_n for all n≥1;n\geq 1; [*]lim⁡n→∞xn=0.\lim_{n\to\infty}x_n=0.
Sequencelimitcollege contestsromania
Romania District MO 2022 Grade 10 P3

Source: Romania District MO 2022 Grade 10

3/28/2022
A positive integer n≥4n\geq 4 is called interesting if there exists a complex number zz such that ∣z∣=1|z|=1 and 1+z+z2+zn−1+zn=0.1+z+z^2+z^{n-1}+z^n=0. Find how many interesting numbers are smaller than 2022.2022.
complex numbersromaniaalgebra
Romania District MO 2022 Grade 12 P3

Source: Romania District MO 2022 Grade 12

3/27/2022
Find all values of n∈N∗n\in\mathbb{N}^* for which In:=∫0πcos⁡(x)⋅cos⁡(2x)⋅…⋅cos⁡(nx) dx=0.I_n:=\int_0^\pi\cos(x)\cdot\cos(2x)\cdot\ldots\cdot\cos(nx) \ dx=0.
Integralcollege contestsromania