5

Part of 2018 Brazil National Olympiad

Problems(2)

Multiples of 7 in the sequence 1, 12, 123, ...

Source: Brazilian Mathematical Olympiad 2018 - Q5

Consider the sequence in which

a_1 = 1

a_n

is obtained by juxtaposing the decimal representation of

n

at the end of the decimal representation of

a_{n-1}

a_1 = 1

a_2 = 12

a_3 = 123

\dots

a_9 = 123456789

a_{10} = 12345678910

and so on. Prove that infinitely many numbers of this sequence are multiples of

7

number theorySequencemultipleBrazilian Math OlympiadBrazilian Math Olympiad 2018

Source: Brazil MO 2018 Grades 8 and 9

One writes, initially, the numbers

1,2,3,\dots,10

in a board. An operation is to delete the numbers

a, b

and write the number

a+b+\frac{ab}{f(a,b)}

f(a, b)

is the sum of all numbers in the board excluding

a

b

, one will make this until remain two numbers

x, y

x\geq y

. Find the maximum value of

x