2

Part of 2010 Pan African

Problems(2)

19 girls in line, no girl is shorter behind/in front of her

Source: Pan African Olympiad 2010

How many ways are there to line up

19

girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?

inductionsearchcombinatorics proposedcombinatorics

Choosing a_0 so that sequence is infinite

Source: Pan African Olympiad 2010

a_0,a_1,a_2,\ldots ,a_n,\ldots

of positive integers is constructed as follows: [*]if the last digit of

a_n

is less than or equal to

5

then this digit is deleted and

a_{n+1}

is the number consisting of the remaining digits. (If

a_{n+1}

contains no digits the process stops.) [*]otherwise

a_{n+1}=9a_n

. Can one choose

a_0

so that an infinite sequence is obtained?

number theory proposednumber theory