1991 AMC 12 第 28 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
28.
一个坛子最初装有 颗黑弹珠和 颗白弹珠。反复从坛中取出三颗弹珠，再按照下表用坛外的弹珠替换：
反复执行这一过程后，坛中可能剩下下列哪一组弹珠？
| 取出的弹珠 | 放回的弹珠 |
|---|---|
| 颗黑弹珠 | 颗黑弹珠 |
| 颗黑弹珠、 颗白弹珠 | 颗黑弹珠、 颗白弹珠 |
| 颗黑弹珠、 颗白弹珠 | 颗白弹珠 |
| 颗白弹珠 | 颗黑弹珠、 颗白弹珠 |
Initially an urn contains black marbles and white marbles. Repeatedly, three marbles are removed from the urn and replaced from a pile outside the urn as follows:
Which of the following sets of marbles could be the contents of the urn after repeated applications of this procedure?
| Marbles removed | Replaced with |
|---|---|
| black | black |
| black, white | black, white |
| black, white | white |
| white | black, white |
颗黑弹珠
black marbles
颗白弹珠
white marbles
颗黑弹珠
black marble
颗黑弹珠和 颗白弹珠
black and white marble
颗白弹珠
white marble
答案：B
小提示：
白弹珠的数量每次只会改变 或
The number of white marbles always changes by or
大提示：
在某次操作后第一次只剩至多两颗弹珠时，检查最后一次操作的可能输出
When an operation first leaves at most two marbles, inspect the possible outputs of that final operation
解答：
白弹珠数量的奇偶性始终不变，所以含一颗白弹珠的选项不可能出现。此外，最后一次操作从三颗弹珠开始时，可能留下 颗黑弹珠、 颗黑弹珠和 颗白弹珠，或 颗白弹珠，但绝不会留下 颗黑弹珠。留下 颗白弹珠的状态确实可以达到：不断把 颗黑弹珠替换为 颗黑弹珠，直到剩下 颗黑弹珠；再两次使用“ 黑 白”的规则，留下 颗白弹珠；然后交替使用“ 白”规则和“ 黑 白”规则，每两次操作把白弹珠数减少 ，直到剩下 颗。
因此正确答案为 B。
The parity of the number of white marbles never changes, so choices with one white marble are impossible. Also a final operation starting with three marbles can leave black, black and white, or white, but never black. The state with white is attainable: repeatedly replace black by black until black remain; twice use the -black--white rule, leaving white; then alternate the -white rule with the -black--white rule, reducing the number of white marbles by per pair until remain.
Thus the correct answer is B.
其他年份的第 28 题
- 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
- 1966 AMC 12
- 1967 AMC 12
- 1968 AMC 12
- 1969 AMC 12
- 1970 AMC 12
- 1971 AMC 12
- 1972 AMC 12
- 1973 AMC 12
- 1974 AMC 12
- 1975 AMC 12
- 1976 AMC 12
- 1977 AMC 12
- 1978 AMC 12
- 1979 AMC 12
- 1980 AMC 12
- 1981 AMC 12
- 1982 AMC 12
- 1983 AMC 12
- 1984 AMC 12
- 1985 AMC 12
- 1986 AMC 12
- 1987 AMC 12
- 1988 AMC 12
- 1989 AMC 12
- 1990 AMC 12
- 1992 AMC 12
- 1993 AMC 12
- 1994 AMC 12
- 1995 AMC 12
- 1996 AMC 12
- 1997 AMC 12
- 1998 AMC 12
- 1999 AMC 12