P1

Part of 2020 Dürer Math Competition (First Round)

Problems(2)

Substitution for numbers

Source: 2019-20 International Dürer Competition, Category E, P1

In maths class Albrecht had to compute

(a+2b-3)^2

. His result was

a^2 +4b^2-9

. ‘This is not correct’ said his teacher, ‘try substituting positive integers for

a

b

.’ Albrecht did so, but his result proved to be correct. What numbers could he substitute? a) Show a good substitution. b) Give all the pairs that Albrecht could substitute and prove that there are no more.

algebrasubstitutions

sums of positive divisors

Source: 2019-20 International Dürer Competition, Category E+, P1

a) Is it possible that the sum of all the positive divisors of two different natural numbers are equal? b) Is it possible that the product of all the positive divisors of two different natural numbers are equal?

number theoryDivisors