MathDB

2

Part of 2012 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2012 Grade 9 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2012

9/25/2018
Let aa, bb, cc, dd, ee, ff and gg be seven distinct positive integers not bigger than 77. Find all primes which can be expressed as abcd+efgabcd+efg
primenumber theory
Regional Olympiad - FBH 2012 Grade 10 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2012

9/25/2018
Harry Potter can do any of the three tricks arbitrary number of times: i)i) switch 11 plum and 11 pear with 22 apples ii)ii) switch 11 pear and 11 apple with 33 plums iii)iii) switch 11 apple and 11 plum with 44 pears In the beginning, Harry had 20122012 of plums, apples and pears, each. Harry did some tricks and now he has 20122012 apples, 20122012 pears and more than 20122012 plums. What is the minimal number of plums he can have?
combinatoricsHarry Potter
Regional Olympiad - FBH 2012 Grade 11 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2012

9/25/2018
On football toornament there were 44 teams participating. Every team played exactly one match with every other team. For the win, winner gets 33 points, while if draw both teams get 11 point. If at the end of tournament every team had different number of points and first place team had 66 points, find the points of other teams
combinatoricsTournament
Regional Olympiad - FBH 2012 Grade 12 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2012

9/25/2018
Let aa, bb, cc and dd be integers such that acac, bdbd and bc+adbc+ad are divisible with positive integer mm. Show that numbers bcbc and adad are divisible with mm
number theoryDivisibility