IMO 1974 SL 1

Problem

I 1 (USA 4)IMO1 Alice, Betty, and Carol took the same series of exam- inations. There was one grade of A, one grade of B, and one grade of C for each examination, where A, B, C are different positive integers. The final test scores were Alice Betty Carol If Betty placed first in the arithmetic examination, who placed second in the spelling examination?

Solution

Denote by n the number of exams. We have n(A+B+C) = 20+10+9 = 39, and since A, B, C are distinct, their sum is at least 6; therefore n = 3 and A + B + C = 13. Assume w.l.o.g. that A > B > C. Since Betty gained A points in arith- metic, but fewer than 13 points in total, she had C points in both remain- ing exams (in spelling as well). Furthermore, Carol also gained fewer than 13 points, but with at least B points on two examinations (on which Betty scored C), including spelling. If she had A in spelling, then she would have at least A + B + C = 13 points in total, a contradiction. Hence, Carol scored B and placed second in spelling. Remark. Moreover, it follows that Alice, Betty, and Carol scored B+A+A, A + C + C, and C + B + B respectively, and that A = 8, B = 4, C = 1.