AMC 8 2026 Problem 10
AMC8 2026 年第 10 题
基本计数原理★★☆☆☆
Five runners completed the grueling Xmarathon: Luke, Melina, Nico, Olympia, and Pedro.Nico finished 11 minutes behind Pedro.Olympia finished 2 minutes ahead of Melina, but 3 minutes behind Pedro.Olympia finished 6 minutes ahead of Luke.
Which runner finished fourth?
五名选手完成了艰苦的 X 马拉松比赛：Luke、Melina、Nico、Olympia 和 Pedro。已知：Nico 比 Pedro 晚了 11 分钟；Olympia 比 Melina 早 2 分钟，但比 Pedro 晚 3 分钟；Olympia 比 Luke 早 6 分钟。
谁是第四名？
- A.Luke
- B.Melina
- C.Nico
- D.Olympia
- E.Pedro
正确答案 · Correct Answer
A
解析 · Solution
解法1
画时间线，以 Pedro 为最前端，逐人放置：
第 1 步：Pedro 第 1。Nico 晚 Pedro 11 分钟，排在 Pedro 后面。
第 2 步：Olympia 晚 Pedro 3 分钟，插到 Pedro 和 Nico 之间。当前顺序：Pedro、Olympia、Nico。
第 3 步：Melina 比 Olympia 晚 2 分钟，插到 Olympia 和 Nico 之间。当前顺序：Pedro、Olympia、Melina、Nico。
第 4 步：Luke 比 Olympia 晚 6 分钟，插到 Melina 和 Nico 之间。
最终顺序：Pedro（第 1）、Olympia（第 2）、Melina（第 3）、Luke（第 4）、Nico（第 5）。
以 Olympia 为基准 O。Pedro 比 Olympia 早 3 分钟 → P = O − 3。Melina 比 Olympia 晚 2 分钟 → M = O + 2。Luke 比 Olympia 晚 6 分钟 → L = O + 6。Nico 比 Pedro 晚 11 分钟，即比 Olympia 晚 8 分钟 → N = O + 8。从快到慢：P < O < M < L < N。第四名是 Luke。
以 Pedro 为基准 P。Olympia = P+3，Melina = P+5，Luke = P+9，Nico = P+11。排序：P < P+3 < P+5 < P+9 < P+11。第四名 = Luke。
Solution 1
Draw a timeline with Pedro at the front. Place each runner one by one:
Step 1: Pedro is 1st. Nico is 11 min behind Pedro, placed behind him.
Step 2: Olympia is 3 min behind Pedro, placed between Pedro and Nico. Order so far: Pedro, Olympia, Nico.
Step 3: Melina is 2 min behind Olympia, placed between Olympia and Nico. Order so far: Pedro, Olympia, Melina, Nico.
Step 4: Luke is 6 min behind Olympia, placed between Melina and Nico.
Final order: Pedro (1st), Olympia (2nd), Melina (3rd), Luke (4th), Nico (5th).
Use Olympia as the reference point. Relative to Olympia (O): Pedro finished 3 min before Olympia, so P = O − 3. Melina finished 2 min after Olympia, so M = O + 2. Luke finished 6 min after Olympia, so L = O + 6. Nico finished 11 min after Pedro = 8 min after Olympia, so N = O + 8. Order from fastest: P < O < M < L < N. The 4th place is Luke.
Use Pedro as the reference point P. Olympia: P+3. Melina: Olympia + 2 = P+5. Luke: Olympia + 6 = P+9. Nico: P+11. Sorted: P < P+3 < P+5 < P+9 < P+11. The 4th place is Luke.
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