Problem Description

This problem set contains multiple fill-in-the-blank tasks. For each task, you only need to compute the result and submit it as required.

Task A: Space

Problem Description

Xiao Lan plans to use $256 \mathrm{MB}$ of memory to create an array. Each element of the array is a $32$-bit binary integer. If we do not consider the space used by the program itself and the extra space needed for memory management, how many $32$-bit binary integers can be stored in $256 \mathrm{MB}$?

Answer Submission

This is a fill-in-the-blank question. 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 make you score $0$.

Task B: Cards

Problem Description

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

Xiao Lan wants to use these cards to form some numbers. He wants to start from $1$ and form positive integers. Each time he forms one number, he keeps it, and the cards used for that number cannot be used to form other numbers.

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

For example, when Xiao Lan has $30$ cards, with $3$ cards for each digit from $0$ to $9$, he can form numbers from $1$ to $10$. But when forming $11$, the digit card $1$ has only one card 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. How far can he form numbers starting from $1$?

Hint: It is recommended to solve this problem using computer programming.

Answer Submission

This is a fill-in-the-blank question. 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 make you score $0$.

Task C: Lines

Problem Description

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

Given $2 \times 3$ integer lattice points on the plane: $\{(x,y) \mid 0 \leq x<2,0 \leq y<3,x \in \mathbb{Z},y \in \mathbb{Z}\}$, that is, 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 $11$ different lines in total.

Given $20 \times 21$ integer lattice points on the plane: $\{(x,y) \mid 0 \leq x<20,0 \leq y<21,x \in \mathbb{Z},y \in \mathbb{Z}\}$, that is, 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 different lines do these points determine in total?

Answer Submission

This is a fill-in-the-blank question. 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 make you score $0$.

Task D: Cargo Placement

Problem Description

Xiao Lan has a huge warehouse and can place many cargos in it.

Now Xiao Lan has $n$ boxes of cargo 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 wants all the cargos to finally form a big cube. 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 different stacking schemes satisfy the requirement?

For example, when $n=4$, there are the following $6$ schemes: $1 \times 1 \times 4$, $1 \times 2 \times 2$, $1 \times 4 \times 1$, $2 \times 1 \times 2$, $2 \times 2 \times 1$, $4 \times 1 \times 1$.

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

Hint: It is recommended to solve this problem using computer programming.

Answer Submission

This is a fill-in-the-blank question. 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 make you score $0$.

Task E: Path

Problem Description

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

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, whose length is 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 $75$.

Please 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 question. 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 make you score $0$.

Input Format

Input one uppercase letter, indicating which task it is.

Output Format

According to the input task label, 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 inside the double quotes with the answer to task A
        "The answer of task B", // Replace inside the double quotes with the answer to task B
        "The answer of task C", // Replace inside the double quotes with the answer to task C
        "The answer of task D", // Replace inside the double quotes with the answer to task D
        "The answer of task E", // Replace inside the double quotes with the answer to task E
    };
    char T;
    cin >> T;
    cout << ans[T - 'A'] << endl;
    return 0;
}

Translated by ChatGPT 5