AMC 8 2024 Problem 19
AMC8 2024 年第 19 题
基本计数原理★★★☆☆
Jordan owns 15 pairs of sneakers.
Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the rest are low-top.
The red high-top sneakers make up a fraction of the collection. What is the least possible value of this fraction?
Jordan 有 15 双运动鞋。其中 的鞋是红色，其余为白色；其中 的鞋是高帮，其余为低帮。红色高帮运动鞋的数量占全部鞋数的一个分数。这个分数的最小可能值是多少？
- A.0
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
C
解析 · Solution
解法1
Jordan 有 10 双高帮运动鞋和 6 双白色运动鞋。我们希望白色高帮鞋尽可能多，所以让 6 双高帮鞋为白色。那么，我们有 10-6=4 双红色高帮鞋，因此答案是
我们首先开始红色和白色运动鞋的数量。 双红色运动鞋，所以有6双白色。然后 双是高帮运动鞋，所以有5双是低帮运动鞋。现在考虑15个位置，前10个标记为高帮运动鞋。如果我们把最后5双作为红色运动鞋插入，则剩下4双红色运动鞋。把这4双作为高帮运动鞋，我们得到答案为C，即 。
有 双红色运动鞋和 6 双白色运动鞋。还有 双高帮运动鞋和 5 双低帮运动鞋。令 为红色高帮运动鞋的数量，令 为白色高帮运动鞋的数量。由此可知，有 9-r 双红色低帮运动鞋和 6-r 双白色低帮运动鞋。我们必须有 才能得到合理的白色运动鞋数量。解这个不等式得到 ，因此 的最小可能值是 4。这意味着会有 9-4=5 双红色低帮运动鞋，所以有 0 双白色低帮运动鞋和 6 双白色高帮运动鞋。这完全符合，因此最小的分数是 。
，这是红色运动鞋的数量。然后， ，所以有十双高帮运动鞋。如果有十双高帮运动鞋，那么就有五双低帮运动鞋。为了使得高帮红色运动鞋的数量最少，所有五双低帮运动鞋都必须是红色的。还剩下四双红色运动鞋，因此它们只能是高帮的。所以答案是 。
设红色高帮运动鞋的数量为 。制作一个双向表格（行：高帮/低帮；列：红/白）：高帮共 10 ，低帮共 5 ，红色共 9 ，白色共 6 ，总计 15 。红色高帮为 ，红色低帮为 9-x 。从表中可以看出 。因此最小值在 x=4 时取得。所以答案是 。
Solution 1
Jordan has 10 high top sneakers, and 6 white sneakers. We would want as many white high sneakers as possible, so we set 6 high sneakers to be white. Then, we have 10-6=4 red high sneakers, so the answer is
We first start the number of red and white sneakers. red sneakers, so 6 are white. Then are high top sneakers, so 5 are low top sneakers. Now think about 15 slots, and the first 10 are labeled high sneakers. If we insert the last 5 sneakers as red sneakers, there are 4 leftover red sneakers. Putting those 4 sneakers as high top sneakers, we have our answer as C, or .
There are red pairs of sneakers and 6 white pairs. There are also high pairs of sneakers and 5 low pairs. Let be the number of red high sneakers and let be the number of white high sneakers. It follows that there are 9-r red pairs of low sneakers and 6-r white pairs. We must have in order to have a valid amount of white sneakers. Solving this inequality gives , so the smallest possible value for is 4. This means that there would be 9-4=5 pairs of low red sneakers, so there are 0 pairs of low white sneakers and 6 pairs of high top white sneakers. This checks out perfectly, so the smallest fraction is .
, which is the number of red pairs of sneakers. Then, , so there are ten pairs of high sneakers. If there are ten high sneakers, there are five low top sneakers. To have the lowest amount of high red sneakers, all five of the low sneakers need to be red. There are four red sneakers remaining so they have to be high So the answer is .
Let the number of red high sneakers be . Making a two way table with rows High/Low and columns Red/White: High total is 10, Low total is 5, Red total is 9, White total is 6, overall 15. The number of red high sneakers is and red low sneakers is 9-x. From the table we can see that . Therefore the minimum value is when x=4. So the answer is .
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