2

Part of 2001 Estonia National Olympiad

Problems(4)

3-digit number wanted

Source: 2001 Estonia National Olympiad Final Round grade 9 p2

Dividing a three-digit number by the number obtained from it by swapping its first and last digit we get

3

as the quotient and the sum of digits of the original number as the remainder. Find all three-digit numbers with this property.

number theoryDigitsremaindersum of digits

min n, among any n integers, there are 3 whose sum is divisible by 3

Source: 2001 Estonia National Olympiad Final Round grade 10 p2

Find the minimum value of

n

such that, among any

n

integers, there are three whose sum is divisible by

3

Sumdivisiblenumber theory

A/B+C/D = E/F , F=lcm(BD), d/E, D/F, prove d/B and d/B

Source: 2001 Estonia National Olympiad Final Round grade 11 p2

A student wrote a correct addition operation

A/B+C/D = E/F

on the blackboard, where both summands are irreducible and

F

is the least common multiple of

B

D

. After that, the student reduced the sum

E/F

correctly by an integer

d

d

is a common divisor of

B

D

least common multipledividesdivisornumber theory

(max) k integers from {1,2... ,2n} so that noone divides another

Source: 2001 Estonia National Olympiad Final Round grade 12 p2

Find the maximum value of

k

for which one can choose

k

integers out of

1,2... ,2n

so that none of them divides another one.

maxnumber theorydivides