Problem Description

Task A: Double Factorial

Problem Description

The double factorial of a positive integer is defined as the product of all positive integers not exceeding it and having the same parity 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 problem 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, enter only this integer; any extra content will not be scored.

Task B: Lattice Points

Problem Description

If a point $(x,y)$ has both coordinates as integers, i.e. $x \in \mathbb{Z}$ and $y \in \mathbb{Z}$, then it is called a lattice point.

If a point $(x,y)$ has both coordinates positive, i.e. $x>0$ and $y>0$, then it lies in the first quadrant.

Among the 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 problem 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, enter only this integer; any extra content will not be scored.

Task C: Integer Partition

Problem Description

Decomposing $3$ into the 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 the sum of three positive integers has $6$ methods: $1+1+3=1+2+2=$ $1+3+1=2+1+2=2+2+1=3+1+1$.

What is the number of methods to decompose $2021$ into the 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, enter only this integer; any extra content will not be scored.

Task D: City-States

Problem Description

Xiaolan Kingdom is a water kingdom with $2021$ city-states, numbered from $1$ to $2021$ in order. Between any two city-states, there is a bridge directly connecting them.

To celebrate a traditional festival, the government of Xiaolan Kingdom plans to decorate some of the bridges.

For two city-states numbered $a$ and $b$, the cost to decorate the bridge between them is calculated as follows: find all digit positions (in base 10) where $a$ and $b$ have different digits, and sum the digits on those positions (from both numbers).

For example, between city-states numbered $2021$ and $922$, the thousands, hundreds, and ones digits are all different. Adding the digits on these positions gives $(2+0+1)+(0+9+2)=14$. Note that $922$ has no thousands digit; treat the thousands digit as $0$.

To save costs, 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 the government must spend to finish the decoration?

Hint: It is recommended to solve this problem 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, enter only this integer; any extra content will not be scored.

Task E: Game

Problem Description

Xiaolan is bored and starts playing a game by himself.

First, a positive integer $n$ is given.

He first writes down a number between $1$ and $n$ on paper. In each subsequent step, Xiaolan may choose a divisor of the last written number (but cannot choose a number that has already been written before) and write it down. This continues until Xiaolan finally writes down $1$.

Xiaolan may have multiple possible plans for the game.

For example, when $n=6$, Xiaolan has $9$ plans: $(1)$, $(2,1),(3,1),(4,1),(4,2,1),(5,1)$, $(6,1),(6,2,1),(6,3,1)$.

When $n=20210509$, how many plans are there?

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, enter only this integer; any extra content will not be scored.

The answer to this problem is relatively large. If you solve it by programming, please use an appropriate data type.

Input Format

Input a capital letter indicating which task it is.

Output Format

According to the input task letter, output the corresponding answer.

Hint

Answer template, for reference.

#include<iostream>
using namespace std;
int main() {
    string ans [] = {
        "The answer of task A", // Replace inside the quotes with the answer for task A
        "The answer of task B", // Replace inside the quotes with the answer for task B
        "The answer of task C", // Replace inside the quotes with the answer for task C
        "The answer of task D", // Replace inside the quotes with the answer for task D
        "The answer of task E", // Replace inside the quotes with the answer for task E
    };
    char T;
    cin >> T;
    cout << ans[T - 'A'] << endl;
    return 0;
}

Translated by ChatGPT 5