1980 AMC 12 Problem 30
Attempt Problem 30 of the 1980 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 1980 AMC 12 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
30.
A six digit number (base ) is squarish if it satisfies the following conditions:
(i) none of its digits is zero;
(ii) it is a perfect square; and
(iii) the first two digits, the middle two digits and the last two digits of the number are all perfect squares when considered as two digit numbers.
How many squarish numbers are there?
Answer: B
Small hint:
List the two-digit perfect squares having no zero digit
Big hint:
Use the first pair to restrict the square root to a short interval, then filter by the final pair
Solution:
Each two-digit block must belong to Restricting the square root by the first block and retaining only squares with an allowed final block leaves the following possible middle blocks:
| first block | possible middle blocks |
|---|---|
Only the middle block is allowed, producing Thus there are squarish numbers.
Therefore, the correct answer is B.
Problem 30 in other years
- 1950 AMC 12
- 1951 AMC 12
- 1952 AMC 12
- 1953 AMC 12
- 1954 AMC 12
- 1955 AMC 12
- 1956 AMC 12
- 1957 AMC 12
- 1958 AMC 12
- 1959 AMC 12
- 1960 AMC 12
- 1961 AMC 12
- 1962 AMC 12
- 1963 AMC 12
- 1964 AMC 12
- 1965 AMC 12
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- 1967 AMC 12
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- 1977 AMC 12
- 1978 AMC 12
- 1979 AMC 12
- 1981 AMC 12
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- 1999 AMC 12