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Print67th Romanian Mathematical Olympiad
Romania algebra
Problem
Each of the numbers is written on a separate card. Alex and Bogdan pick cards each and notice that the sum of Alex's cards is three times larger than the sum of Bogdan's cards. Find the number of the remaining card.
Solution
Let and be the respective sums. Then is at least , so is at least . But is at most , so must be and must be . Therefore Alex must pick and Bogdan has to pick , so the remaining card must have written on it.
Final answer
16
Techniques
Sums and productsColoring schemes, extremal arguments