Problem Description

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 Format

Input a single uppercase letter indicating which task it is.

Output Format

Output the answer corresponding to the given task letter.

Hint

Answer template for reference.

#include<iostream>
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 << ans[T - 'A'] << endl;
    return 0;
}

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