1965 AMC 12 Problem 29
Attempt Problem 29 of the 1965 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1965 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
29.
Of students taking at least one subject the number taking Mathematics and English only equals the number taking Mathematics only. No student takes English only or History only, and six students take Mathematics and History, but no English. The number taking English and History only is five times the number taking all three subjects. If the number taking all three subjects is even and non-zero, the number taking English and Mathematics only is:
Answer: A
Small Hint:
Let be the all-three count and both the Math-only and Math-English-only count
Big Hint:
The total becomes
Solution:
Let be the number taking all three and the number taking Mathematics and English only, which also equals the Mathematics-only count. The disjoint-region total is so Since is positive and even, is the only value giving a nonnegative listed count, and
Therefore, the correct answer is A.
Problem 29 in Other Years
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