MathDB

Problem 2

Part of 2001 Moldova National Olympiad

Problems(6)

n+s(n)=2004, sum of digits=s(n) (2001 Moldova MO Grade 7 P2)

Source:

4/12/2021
Let S(n)S(n) denote the sum of digits of a natural number nn. Find all nn for which n+S(n)=2004n+S(n)=2004.
number theory
Maximum of min(a1-a1a2,a2-a2a3,...) where a_n in [0,1]

Source: 2001 Moldova MO Grade 8 P2

4/12/2021
If n∈Nn\in\mathbb N and a1,a2,…,ana_1,a_2,\ldots,a_n are arbitrary numbers in the interval [0,1][0,1], find the maximum possible value of the smallest among the numbers a1−a1a2,a2−a2a3,…,an−ana1a_1-a_1a_2,a_2-a_2a_3,\ldots,a_n-a_na_1.
Inequalityoptimizationinequalities
sum of two consecutive primes never product of two primes

Source: 2001 Moldova MO Grade 9 P2

4/12/2021
Prove that the sum of two consecutive prime numbers is never a product of two prime numbers.
number theory
product=sum of 2003 odd naturals

Source: 2001 Moldova MO Grade 10 P2

4/13/2021
Prove that there are no 20032003 odd positive integers whose product equals their sum. Is the previous proposition true for 20012001 odd positive integers?
number theory
product from vertex to other vertices in n-gon

Source: 2001 Moldova MO Grade 12 P2

4/13/2021
A regular nn-gon is inscribed in a unit circle. Compute the product from a fixed vertex to all the other vertices.
geometry
find limit of a sequence with recurrence relation

Source: 2001 Moldova MO Grade 11 P2

4/13/2021
Let m≥2m\ge2 be an integer. The sequence (an)n∈N(a_n)_{n\in\mathbb N} is defined by a0=0a_0=0 and an=⌊nm⌋+a⌊nm⌋a_n=\left\lfloor\frac nm\right\rfloor+a_{\left\lfloor\frac nm\right\rfloor} for all nn. Determine lim⁡n→∞ann\lim_{n\to\infty}\frac{a_n}n.
recurrence relationSequencealgebra