AMC10 2013 B

AMC10 2013 B · Q17

AMC10 2013 B · Q17. It mainly tests Linear equations, Invariants.

Alex has 75 red tokens and 75 blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token, and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?

Alex 有 75 个红色代币和 75 个蓝色代币。有一个摊位，Alex 可以交出两个红色代币，换取一个银色代币和一个蓝色代币；另一个摊位，Alex 可以交出三个蓝色代币，换取一个银色代币和一个红色代币。Alex 继续交换代币直到无法再交换。最终 Alex 会有多少银色代币？

(A) 62 62

(B) 82 82

(D) 102 102

(E) 103 103

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): After Alex makes $m$ exchanges at the first booth and $n$ exchanges at the second booth, Alex has $75-(2m-n)$ red tokens, $75-(3n-m)$ blue tokens, and $m+n$ silver tokens. No more exchanges are possible when he has fewer than $2$ red tokens and fewer than $3$ blue tokens. Therefore no more exchanges are possible if and only if $2m-n\ge 74$ and $3n-m\ge 73$. Equality can be achieved when $(m,n)=(59,44)$, and Alex will have $59+44=103$ silver tokens. Note that the following exchanges produce 103 silver tokens: \[ \begin{array}{|l|c|c|c|} \hline & \text{Red Tokens} & \text{Blue Tokens} & \text{Silver Tokens} \\ \hline \text{Exchange 75 blue tokens} & 100 & 0 & 25 \\ \hline \text{Exchange 100 red tokens} & 0 & 50 & 75 \\ \hline \text{Exchange 48 blue tokens} & 16 & 2 & 91 \\ \hline \text{Exchange 16 red tokens} & 0 & 10 & 99 \\ \hline \text{Exchange 9 blue tokens} & 3 & 1 & 102 \\ \hline \text{Exchange 2 red tokens} & 1 & 2 & 103 \\ \hline \end{array} \]

答案（E）：当 Alex 在第一个摊位进行了 $m$ 次交换、在第二个摊位进行了 $n$ 次交换后，Alex 有 $75-(2m-n)$ 个红色代币、$75-(3n-m)$ 个蓝色代币，以及 $m+n$ 个银色代币。当他拥有少于 $2$ 个红色代币且少于 $3$ 个蓝色代币时，就无法再进行交换。因此，无法再交换当且仅当 $2m-n\ge 74$ 且 $3n-m\ge 73$。当 $(m,n)=(59,44)$ 时可取到等号，此时 Alex 将有 $59+44=103$ 个银色代币。注意：下面这些交换会得到 103 个银色代币： \[ \begin{array}{|l|c|c|c|} \hline & \text{红色代币} & \text{蓝色代币} & \text{银色代币} \\ \hline \text{交换 75 个蓝色代币} & 100 & 0 & 25 \\ \hline \text{交换 100 个红色代币} & 0 & 50 & 75 \\ \hline \text{交换 48 个蓝色代币} & 16 & 2 & 91 \\ \hline \text{交换 16 个红色代币} & 0 & 10 & 99 \\ \hline \text{交换 9 个蓝色代币} & 3 & 1 & 102 \\ \hline \text{交换 2 个红色代币} & 1 & 2 & 103 \\ \hline \end{array} \]

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