Problem Description

Task A: Cards

Problem Description

Xiao Lan has many digit cards. Each card has a digit from $0$ to $9$ on it.

Xiao Lan plans to use these cards to form some numbers. He wants to form positive integers starting from $1$. After forming a number, he keeps it, and the cards used for it cannot be used to form other numbers.

Xiao Lan wants to know how far he can form numbers starting from $1$.

For example, when Xiao Lan has $30$ cards, with $3$ cards for each digit from $0$ to $9$, Xiao Lan can form $1$ to $10$. But when forming $11$, he has only one card with digit $1$ left, which is not enough to form $11$.

Now Xiao Lan has $2021$ cards for each digit from $0$ to $9$, a total of $20210$ cards. Up to what number can Xiao Lan form starting from $1$?

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 of this problem is an integer. When submitting, only fill in this integer. Any extra content will result in no score.

Task B: Lines

Problem Description

In the Cartesian coordinate system, two points determine a line. If multiple points lie on the same line, then the line determined by any two of these points is the same line.

Given $2 \times 3$ lattice points on the plane $\{(x,y) \mid 0 \leq x<2,0 \leq y<3,x \in \mathbb{Z},y \in \mathbb{Z}\}$, i.e., points whose $x$-coordinate is an integer between $0$ and $1$ (including $0$ and $1$), and whose $y$-coordinate is an integer between $0$ and $2$ (including $0$ and $2$). These points determine $11$ distinct lines in total.

Given $20 \times 21$ lattice points on the plane $\{(x,y) \mid 0 \leq x<20,0 \leq y<21,x \in \mathbb{Z},y \in \mathbb{Z}\}$, i.e., points whose $x$-coordinate is an integer between $0$ and $19$ (including $0$ and $19$), and whose $y$-coordinate is an integer between $0$ and $20$ (including $0$ and $20$). How many distinct lines do these points determine in total?

Answer Submission

This is a 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, only fill in this integer. Any extra content will result in no score.

Task C: Cargo Placement

Problem Description

Xiao Lan has a super large warehouse that can store many goods.

Now, Xiao Lan has $n$ boxes of goods to place in the warehouse. Each box is a regular cube. Xiao Lan defines three mutually perpendicular directions: length, width, and height. The edges of each box must be strictly parallel to the length, width, and height directions.

Xiao Lan hopes that all the goods will finally be stacked into one large cuboid. That is, stack $L$, $W$, and $H$ boxes along the length, width, and height directions respectively, satisfying $n=L \times W \times H$.

Given $n$, how many stacking plans meet the requirement?

For example, when $n=4$, there are $6$ plans: $1 \times 1 \times 4 、 1 \times 2 \times 2 、 1 \times 4 \times 1 、 2 \times 1 \times 2$ 、 $2 \times 2 \times 1 、 4 \times 1 \times 1$.

When $n=2021041820210418$ (note that it has $16$ digits), how many plans are there in total?

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 of this problem is an integer. When submitting, only fill in this integer. Any extra content will result in no score.

Task D: Path

Problem Description

After learning shortest paths, Xiao Lan is very happy. He defined a special graph and wants to find the shortest path in this graph.

Xiao Lan's graph consists of $2021$ nodes, numbered from $1$ to $2021$ in order.

For two different nodes $a,b$, if the absolute value of the difference between $a$ and $b$ is greater than $21$, then there is no edge between them. If the absolute value of the difference between $a$ and $b$ is less than or equal to $21$, then there is an undirected edge between them with length equal to the least common multiple of $a$ and $b$.

For example, there is no edge between node $1$ and node $23$. There is an undirected edge between node $3$ and node $24$ with length $24$. There is an undirected edge between node $15$ and node $25$ with length $75$.

Compute the length of the shortest path between node $1$ and node $2021$.

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 of this problem is an integer. When submitting, only fill in this integer. Any extra content will result in no score.

Task E: Cycle Counting

Problem Description

Lanqiao Academy has $21$ teaching buildings, numbered from $1$ to $21$. For two buildings $a$ and $b$, when $a$ and $b$ are coprime, there is a corridor directly connecting them, and it can be traveled in both directions. Otherwise, there is no corridor directly connecting them.

Xiao Lan is now in building $1$. He wants to visit each building exactly once and finally return to building $1$ (i.e., walk along a Hamiltonian cycle). How many different visiting plans does he have? Two visiting plans are considered different if there exists some $i$ such that after visiting building $i$, Xiao Lan visits different next buildings in the two plans.

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 of this problem is an integer. When submitting, only fill in this integer. Any extra content will result in no score.

Input Format

Input one uppercase letter, indicating which task it is.

Output Format

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

Hint

Answer template for reference.

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

Translated by ChatGPT 5