AMC10 2024 B

AMC10 2024 B · Q16

AMC10 2024 B · Q16. It mainly tests Invariants, Parity (odd/even).

Jerry likes to play with numbers. One day, he wrote all the integers from $1$ to $2024$ on the whiteboard. Then he repeatedly chose four numbers on the whiteboard, erased them, and replaced them by either their sum or their product. (For example, Jerry's first step might have been to erase $1$, $2$, $3$, and $5$, and then write either $11$, their sum, or $30$, their product, on the whiteboard.) After repeatedly performing this operation, Jerry noticed that all the remaining numbers on the whiteboard were odd. What is the maximum possible number of integers on the whiteboard at that time?

杰瑞喜欢玩数字。有一天，他在白板上写下了从$1$到$2024$的所有整数。然后他反复选择白板上的四个数字，擦掉它们，并用它们的和或积替换它们。（例如，杰瑞的第一步可能是擦掉$1$、$2$、$3$和$5$，然后写下它们的和$11$或积$30$。）在反复进行这个操作后，杰瑞注意到白板上剩余的所有数字都是奇数。当时白板上可能的最大整数个数是多少？

(A) 1010 1010

(B) 1011 1011

(D) 1013 1013

(E) 1014 1014

Answer

Correct choice: (A)

正确答案：（A）

Solution

Consider the numbers as $1,0,1,0,...,1,0$. Note that the number of odd integers is monotonously decreasing. We need to get rid of $1012$ $0$'s, so we either add or multiply four $0$s together to get $1012\rightarrow 253 \rightarrow 63+1=64 \rightarrow 16 \rightarrow 4 \rightarrow 1.$ To get rid of the final $0$, we need to consume three other $1$'s to result in one $1$. Thus the answer is $1012-2=\boxed{\textbf{(A) } 1010 }$,

将数字视为$1,0,1,0,...,1,0$。注意奇数个数是单调递减的。我们需要消除$1012$个$0$，所以我们将四个$0$相加或相乘来得到$1012\rightarrow 253 \rightarrow 63+1=64 \rightarrow 16 \rightarrow 4 \rightarrow 1$。要消除最后一个$0$，我们需要消耗另外三个$1$来得到一个$1$。因此答案是$1012-2=\boxed{\textbf{(A) } 1010 }$。

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