PUMaC 2014 · 代数（A 组） · 第 2 题

PUMaC 2014 — Algebra (Division A) — Problem 2

[ 3 ] Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price p . A day later, the U-Store ′ ′ decides to give a discount on plushies and sell them at p with 0 < p < p . Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has 12 plushies, Bob has 40, and Charlie has 52 but they all spent the same amount of money:$42. How many plushies did Alice buy on the first day?

解析

[ 3 ] Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price p . A day later, the ′ ′ U-Store decides to give a discount on plushies and sell them at p with 0 < p < p . Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has 12 plushies, Bob has 40, and Charlie has 52 but they all spent the same amount of money:$42. How many plushies did Alice buy on the first day? Solution: If we let a, b, c denote the of plushies Alice, Bob, and Charlie bought the first day, then the next day they bought 12 − a, 40 − b, 52 − c more plushies respectively. So, we have the set of equations: ′ ap + (12 − a ) p = 42 ′ bp + (40 − b ) p = 42 ′ cp + (52 − c ) p = 42 ⇓ ′ ′ a ( p − p ) + 12 p = 42 ′ ′ b ( p − p ) + 40 p = 42 ′ ′ c ( p − p ) + 52 p = 42 ′ ′ Then subtracting the second equation from the first, we get that ( a − b )( p − p ) = 28 p ′ ′ and subtracting the third equation from the second we get ( c − b )( p − p ) = 12 p and so a − b c − b a − b c − b = ⇒ = . Since 1 ≤ a < 12 we must have that a − b = 7 and c − b = 3 28 12 7 3 and so c − a = 10. Since c ≥ 1 and a < 12, we must have a = 11 , b = 4 , c = 1.