P8751 [Lanqiao Cup 2021 NOI Qualifier B2] Fill-in-the-Blank Problems

This problem consists of five fill-in-the-blank tasks, A to E. You only need to compute the result and submit it. Each result is an integer. When submitting the answer, write only that integer; any extra content will not receive points. ## Task A: Modulo ### Problem Description In languages such as $\mathrm{C} / \mathrm{C}++/ \mathrm{Java} / \mathrm{Python}$, `%` denotes the modulo operation. What is the value of `2021%20`? ### Answer Submission This is a fill-in-the-blank problem. You only need to compute the result and submit it. The result is an integer. When submitting, write only that integer; any extra content will not receive points. ## Task B: Double Factorial ### Problem Description The double factorial of a positive integer is defined as the product of all positive integers not exceeding it that have the same parity (odd/even) as it. The double factorial of $n$ is denoted by $n!!$. For example: $3!!=3 \times 1=3$. $8!!=8 \times 6 \times 4 \times 2=384$. $11!!=11 \times 9 \times 7 \times 5 \times 3 \times 1=10395$. What are the last $5$ digits (in decimal) of $2021!!$? Note: $2021!!=2021 \times 2019 \times \cdots \times 5 \times 3 \times 1$. Hint: It is recommended to solve this using computer programming. ### Answer Submission This is a fill-in-the-blank problem. You only need to compute the result and submit it. The result is an integer. When submitting, write only that integer; any extra content will not receive points. ## Task C: Lattice Points ### Problem Description If both coordinates of a point $(x,y)$ are integers, i.e., $x \in \mathbb{Z}$ and $y \in \mathbb{Z}$, then the point is called a lattice point. If both coordinates of a point $(x,y)$ are positive, i.e., $x>0$ and $y>0$, then the point lies in the first quadrant. Among lattice points in the first quadrant, how many points $(x,y)$ satisfy that the product of the two coordinates is at most $2021$, i.e., $x \cdot y \leq 2021$? Hint: It is recommended to solve this using computer programming. ### Answer Submission This is a fill-in-the-blank problem. You only need to compute the result and submit it. The result is an integer. When submitting, write only that integer; any extra content will not receive points. ## Task D: Integer Decomposition ### Problem Description Decomposing $3$ into a sum of two positive integers has two methods: $3=1+2$ and $3=2+1$. Note that different orders count as different methods. Decomposing $5$ into a sum of three positive integers has $6$ methods: $1+1+3$, $1+2+2$, $1+3+1$, $2+1+2$, $2+2+1$, $3+1+1$. How many methods are there to decompose $2021$ into a sum of five positive integers? ### Answer Submission This is a fill-in-the-blank problem. You only need to compute the result and submit it. The result is an integer. When submitting, write only that integer; any extra content will not receive points. ## Task E: City-States ### Problem Description The country of Xiaolan is a water kingdom with $2021$ city-states, numbered from $1$ to $2021$. Between every pair of city-states, there is a bridge directly connecting them. To celebrate a traditional festival, the government of Xiaolan plans to decorate some of the bridges. For two city-states numbered $a$ and $b$, the cost to decorate the bridge between them is computed as follows: find all digit positions (in base 10) where $a$ and $b$ have different digits, and sum the digits at those positions from both numbers. For example, for city-states $2021$ and $922$, the thousands, hundreds, and ones digits are all different. Summing the digits at these positions gives $(2+0+1)+(0+9+2)=14$. Note that $922$ has no thousands digit, so its thousands digit is treated as $0$. To save money, the government plans to decorate only $2020$ bridges, and it must be guaranteed that from any city-state to any other city-state, one can travel using only decorated bridges. What is the minimum total cost needed to complete the decoration? Hint: It is recommended to solve this using computer programming. ### Answer Submission This is a fill-in-the-blank problem. You only need to compute the result and submit it. The result is an integer. When submitting, write only that integer; any extra content will not receive points.

Input a single uppercase letter indicating which task it is.

Output the answer corresponding to the given task letter.

Answer template for reference. ```cpp #include using namespace std; int main() { string ans [] = { "The answer of task A", // Replace inside quotes with the answer to task A "The answer of task B", // Replace inside quotes with the answer to task B "The answer of task C", // Replace inside quotes with the answer to task C "The answer of task D", // Replace inside quotes with the answer to task D "The answer of task E", // Replace inside quotes with the answer to task E }; char T; cin >> T; cout