MathDB

4

Part of 2015 Bosnia And Herzegovina - Regional Olympiad

Problems(3)

Regional Olympiad - FBH 2015 Grade 10 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2015

9/23/2018
On competition there were 6767 students. They were solving 66 problems. Student who solves kkth problem gets kk points, while student who solves incorrectly kkth problem gets −k-k points. a)a) Prove that there exist two students with exactly the same answers to problems b)b) Prove that there exist at least 44 students with same number of points
combinatorics
Regional Olympiad - FBH 2015 Grade 9 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2015

9/23/2018
Alice and Mary were searching attic and found scale and box with weights. When they sorted weights by mass, they found out there exist 55 different groups of weights. Playing with the scale and weights, they discovered that if they put any two weights on the left side of scale, they can find other two weights and put on to the right side of scale so scale is in balance. Find the minimal number of weights in the box
combinatoricsSets
Regional Olympiad - FBH 2015 Grade 12 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2015

9/23/2018
It is given set A={1,2,3,...,2n−1}A=\{1,2,3,...,2n-1\}. From set AA, at least n−1n-1 numbers are expelled such that: a)a) if number a∈Aa \in A is expelled, and if 2a∈A2a \in A then 2a2a must be expelled b)b) if a,b∈Aa,b \in A are expelled, and a+b∈Aa+b \in A then a+ba+b must be also expelled Which numbers must be expelled such that sum of numbers remaining in set stays minimal
combinatoricsSets