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Problem 40

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Champernowne’s constant

An irrational decimal fraction is created by concatenating the positive integers:

$$0.12345678910{\color{red}1}112131415161718192021…$$

It can be seen that the $12$^th digit of the fractional part is $1$.

If $d_n$ represents the $n$^th digit of the fractional part, find the value of the following expression.

$$d_1 \times d_{10} \times d_{100} \times d_{1000} \times d_{10000} \times d_{100000} \times d_{1000000}$$

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钱珀瑙恩常数

将所有正整数连接起来构造的一个十进制无理数如下所示：

$$0.12345678910{\color{red}1}112131415161718192021…$$

可以看出小数点后第$12$位数字是$1$。

如果$d_n$表示上述无理数小数点后的第$n$位数字，求下式的值：

$$d_1 \times d_{10} \times d_{100} \times d_{1000} \times d_{10000} \times d_{100000} \times d_{1000000}$$

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