1. I 1 (USA 4) ${ }^{\mathrm{IMO}}$ Alice, Betty, and Carol took the same series of examinations. 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 | | :---: | :---: | :---: | | 20 | 10 | 9 |  If Betty placed first in the arithmetic examination, who placed second in the spelling examination?
1. 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 arithmetic, but fewer than 13 points in total, she had \( C \) points in both remaining 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 \( A = 8, B = 4, C = 1 \).