MathDB

4

Part of 2019 Middle European Mathematical Olympiad

Problems(2)

smallest no. of consecutive numbers

Source: 2019 MEMO Problem I-4

8/29/2019
Determine the smallest positive integer nn for which the following statement holds true: From any nn consecutive integers one can select a non-empty set of consecutive integers such that their sum is divisible by 20192019.
Proposed by Kartal Nagy, Hungary
number theorymemoMEMO 2019
2019 no's as an arithmetic expression

Source: 2019 MEMO Problem T-4

8/30/2019
Prove that every integer from 11 to 20192019 can be represented as an arithmetic expression consisting of up to 1717 symbols 22 and an arbitrary number of additions, subtractions, multiplications, divisions and brackets. The 22's may not be used for any other operation, for example, to form multidigit numbers (such as 222222) or powers (such as 222^2).
Valid examples: ((2×2+2)×2−22)×2=22    ,    (2×2×2−2)×(2×2+2+2+22)=42\left((2\times 2+2)\times 2-\frac{2}{2}\right)\times 2=22 \;\;, \;\; (2\times2\times 2-2)\times \left(2\times 2 +\frac{2+2+2}{2}\right)=42
Proposed by Stephan Wagner, Austria
number theorycombinatoricsmemoMEMO 2019