Question

Ed 和 Sue 骑自行车的速度相同且恒定。同样地，他们慢跑的速度相同且恒定，游泳的速度也相同且恒定。 Ed 骑自行车 $2$ 小时、慢跑 $3$ 小时、游泳 $4$ 小时后共行进 $74$ 千米； Sue 慢跑 $2$ 小时、游泳 $3$ 小时、骑自行车 $4$ 小时后共行进 $91$ 千米。他们骑自行车、慢跑和游泳的速度都是每小时整数千米。求 Ed 骑自行车、慢跑和游泳速度的平方和。

Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $74$ kilometers after biking for $2$ hours, jogging for $3$ hours, and swimming for $4$ hours, while Sue covers $91$ kilometers after jogging for $2$ hours, swimming for $3$ hours, and biking for $4$ hours. Their biking, jogging, and swimming rates are all whole numbers of kilometers per hour. Find the sum of the squares of Ed's biking, jogging, and swimming rates.

答案

答案：314

知识点：方程组丢番图方程速率

难度评级：2020

解答：

设 $b$ 、 $j$ 、 $s$ 分别为骑自行车、慢跑和游泳的速度。两次行程给出 $2b + 3j + 4s = 74$ 和 $4b + 2j + 3s = 91.$ 将第一个方程乘以两倍后减去第二个方程，得 $4j + 5s = 57$ ，其正整数解为 $(j, s) = (13, 1)$ 、 $(8, 5)$ 和 $(3, 9)$ 。

对应的 $2b = 74 - 3j - 4s$ 的值分别为 $31$ 、 $30$ ，和 $29$ ，所以只有 $(j, s) = (8, 5)$ 给出整数速度 $b = 15$ 。平方和为 $15^2 + 8^2 + 5^2$ $= 225 + 64 + 25$ $= 314$ 。

Let $b,$ $j,$ and $s$ be the biking, jogging, and swimming rates. The two trips give $2b + 3j + 4s = 74$ and $4b + 2j + 3s = 91.$ Doubling the first equation and subtracting the second yields $4j + 5s = 57,$ whose positive integer solutions are $(j, s) = (13, 1),$ $(8, 5),$ and $(3, 9).$

The corresponding values of $2b = 74 - 3j - 4s$ are $31,$ $30,$ and $29,$ so only $(j, s) = (8, 5)$ gives a whole-number rate, $b = 15.$ The sum of the squares is $15^2 + 8^2 + 5^2$ $= 225 + 64 + 25$ $= 314.$

2008 AIME I 第 3 题

答案：314

难度评级：2020

解答：

其他年份的第 3 题