3

Part of 2005 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Integer divisible by numbers obtained by deleting its digits

Source: XIV Rioplatense Mathematical Olympiad (2005), Level 3

Find the largest positive integer

n

not divisible by

10

which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of

n

n

is written in the usual base ten notation.)

number theory unsolvednumber theory

rioplatense(2005/6)

Source:

k

be a positive integer. Show that for all

n>k

there exist convex figures

F_{1},\ldots, F_{n}

F

such that there doesn't exist a subset of

k

F_{1},..., F_{n}

F

is covered for this elements, but

F

is covered for every subset of

k+1

F_{1}, F_{2},....., F_{n}

integrationcombinatorics unsolvedcombinatorics