MathDB

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Part of 2017 Bundeswettbewerb Mathematik

Problems(2)

Removing Numbers On A Blackboard

Source: Bundeswettbewerb Mathematik 2017, Round 1 - #1

8/7/2017
The numbers 1,2,3,…,20171,2,3,\dots,2017 are on the blackboard. Amelie and Boris take turns removing one of those until only two numbers remain on the board. Amelie starts. If the sum of the last two numbers is divisible by 88, then Amelie wins. Else Boris wins. Who can force a victory?
blackboardcombinatoricscombinatorics unsolvednumber theorynumber theory unsolveddivisibleDivisibility
Pairwise Distinct Differences

Source: Bundeswettbewerb Mathematik 2017, Round 2 - #1

9/3/2017
For which integers n≥4n \geq 4 is the following procedure possible? Remove one number of the integers 1,2,3,…,n+11,2,3,\dots,n+1 and arrange them in a sequence a1,a2,…,ana_1,a_2,\dots,a_n such that of the nn numbers ∣a1−a2∣,∣a2−a3∣,…,∣an−1−an∣,∣an−a1∣ |a_1-a_2|,|a_2-a_3|,\dots,|a_{n-1}-a_n|,|a_n-a_1| no two are equal.
combinatoricscombinatorics unsolvedSequencebundeswettbewerb