5

Part of 2016 Tuymaada Olympiad

Problems(2)

Two prime numbers

Source: Tuymaada 2016, P5 . Senior League

The ratio of prime numbers

p

q

does not exceed 2 (

p\ne q

). Prove that there are two consecutive positive integers such that the largest prime divisor of one of them is

p

and that of the other is

q

number theorynumber theory proposedTuymaadaprime numbers

Source:

Positive numbers are written in the squares of a 10 × 10 table. Frogs sit in five squares and cover the numbers in these squares. Kostya found the sum of all visible numbers and got 10. Then each frog jumped to an adjacent square and Kostya’s sum changed to

10^2

. Then the frogs jumped again, and the sum changed to

10^3

and so on: every new sum was 10 times greater than the previous one. What maximum sum can Kostya obtain?

combinatoricsTuymaada