1

Part of 2007 France Team Selection Test

Problems(2)

France TST 2007

Source: Problem 1

For a positive integer

a

a'

is the integer obtained by the following method: the decimal writing of

a'

is the inverse of the decimal writing of

a

(the decimal writing of

a'

can begin by zeros, but not the one of

a

); for instance if

a=2370

a'=0732

732

a_{1}

be a positive integer, and

(a_{n})_{n \geq 1}

the sequence defined by

a_{1}

and the following formula for

n \geq 1

a_{n+1}=a_{n}+a'_{n}.

a_{7}

number theory proposednumber theory

France TST 2007

Source: Problem 4

5

points in the space, such that for all

n\in\{1,2,\ldots,10\}

there exist two of them at distance between them

n

combinatorics proposedcombinatorics