Browse · MathNet
PrintTHE 68th ROMANIAN MATHEMATICAL OLYMPIAD
Romania counting and probability
Problem
Consider the set of positive 3-digit integers consisting of three consecutive, not necessarily ordered, digits.
a) Determine the cardinal of the set .
b) Prove that for any choice of some elements of the set , the sum of the chosen elements cannot equal 2017.
a) Determine the cardinal of the set .
b) Prove that for any choice of some elements of the set , the sum of the chosen elements cannot equal 2017.
Solution
a) generate 4 numbers. Each of the triplets , , ..., generate 6 numbers. The cardinal of the set is .
b) Any number in is a multiple of 3. 2017 being not a multiple of 3, the sum of any numbers in cannot be 2017.
b) Any number in is a multiple of 3. 2017 being not a multiple of 3, the sum of any numbers in cannot be 2017.
Final answer
46; no sum of chosen elements can equal 2017
Techniques
Enumeration with symmetryDivisibility / Factorization