MathDB

Problem 4

Part of 1998 Croatia National Olympiad

Problems(4)

(n+1)^m-n and (n+1)^(m+3)-n

Source: Croatia 1998 2nd Grade P4

6/7/2021
For natural numbers m,nm,n, set a=(n+1)m−na=(n+1)^m-n and b=(n+1)m+3−nb=(n+1)^{m+3}-n. (a) Prove that aa and bb are coprime if mm is not divisible by 33. (b) Find all numbers m,nm,n for which aa and bb are not coprime.
number theory
hexagon, circle with sides as diameters

Source: Croatia 1998 1st Grade P4

6/7/2021
Let there be given a regular hexagon of side length 11. Six circles with the sides of the hexagon as diameters are drawn. Find the area of the part of the hexagon lying outside all the circles.
geometryhexagonpolygoncircles
NT PHP, 13|sum of digits

Source: Croatia 1998 3rd Grade P4

6/8/2021
Among any 7979 consecutive natural numbers there exists one whose sum of digits is divisible by 1313. Find a sequence of 7878 consecutive natural numbers for which the above statement fails.
Phpnumber theory
switching lightbulbs in different sattes

Source: Croatia 1998 4th Grade P4

6/8/2021
Eight bulbs are arranged on a circle. In every step we perform the following operation: We simultaneously switch off all those bulbs whose two neighboring bulbs are in different states, and switch on the other bulbs. Prove that after at most four steps all the bulbs will be switched on.
combinatorics