P16783 [Lanqiao Cup 2026 National A] Key Extraction

Xiao Lan is doing a security evaluation for an encryption algorithm based on big integer multiplication. During the evaluation, he uses this algorithm to generate two special keys $A$ and $B$: - Key $A$ consists of $20260606$ digits $9$, that is, $A = \underbrace{99...9}_{20260606\text{ digits}}$. - Key $B$ consists of $2026$ digits $9$, that is, $B = \underbrace{99...9}_{2026\text{ digits}}$. By computing their product, Xiao Lan gets an extremely long combined key $S = A \times B$. To verify the integrity of this key during transmission, Xiao Lan needs to extract the digits at specific positions in the decimal representation of $S$ (counting from left to right, with no leading zeros). Let $d_1, d_2, d_3$ be the digits at the $2026$-th, $20260523$-rd, and $20262632$-nd positions of $S$, respectively. Now, please find the value of the three-digit integer $\overline{d_1 d_2 d_3}$ formed by these three extracted digits in order. Here $d_1$ is the hundreds digit (it is guaranteed that $d_1 \ne 0$), $d_2$ is the tens digit, and $d_3$ is the ones digit. For example, if the three extracted digits are $d_1 = 2, d_2 = 5, d_3 = 7$, then the answer you should submit is $257$.

N/A

This is an output-only (fill-in-the-blank) problem. You only need to compute the result and submit it. The result of this problem is an integer. When submitting, you only need to write a program that outputs this integer. Any extra content will result in no score.

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