Question

Bill 开始刷漆前有 $130$ 盎司蓝漆、 $164$ 盎司红漆和 $188$ 盎司白漆。Bill 在墙上刷了四条等宽的条纹：一条蓝色、一条红色、一条白色和一条粉色。粉色由红色和白色混合而成，比例不一定相等。刷完后，Bill 剩下的蓝漆、红漆和白漆量相等。求 Bill 一共剩下多少盎司油漆。

Before starting to paint, Bill had $130$ ounces of blue paint, $164$ ounces of red paint, and $188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left.

答案

答案：114

知识点：一次方程代数变形

难度评级：1750

解答：

设每条条纹用 $s$ 盎司油漆。蓝漆只用于蓝色条纹，所以用了 $s$ 盎司蓝漆。由于三种颜色的剩余量相等，而红漆和白漆一开始分别比蓝漆多 $34$ 和 $58$ 盎司，所以红漆用量比蓝漆多 $164 - 130 = 34$ 盎司，白漆用量比蓝漆多 $188 - 130 = 58$ 盎司。多出的这些红漆和白漆正好组成粉色条纹，因此 $s = 34 + 58 = 92$ 。

因此 Bill 每种颜色都剩下 $130 - 92 = 38$ 盎司，总共剩下 $3 \cdot 38 = 114$ 盎司。

Say each stripe used $s$ ounces of paint. Blue was used only on the blue stripe, so $s$ ounces of blue were used. Since the three leftovers are equal and the colors started $34$ and $58$ ounces apart, red use exceeded blue use by $164 - 130 = 34$ ounces and white use exceeded blue use by $188 - 130 = 58$ ounces. That extra red and white is exactly the pink stripe, so $s = 34 + 58 = 92.$

Bill therefore had $130 - 92 = 38$ ounces of each color left, for a total of $3 \cdot 38 = 114$ ounces.

2009 AIME II 第 1 题

答案：114

难度评级：1750

解答：

其他年份的第 1 题