A14019 | Ceil Divisions
时间限制1s
内存限制256MB
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题目描述
You have an array $a_1, a_2, \dots, a_n$ where $a_i = i$ .
In one step, you can choose two indices $x$ and $y$ ( $x \neq y$ ) and set $a_x = \left\lceil \frac{a_x}{a_y} \right\rceil$ (ceiling function).
Your goal is to make array $a$ consist of $n - 1$ ones and $1$ two in no more than $n + 5$ steps. Note that you don't have to minimize the number of steps.
In one step, you can choose two indices $x$ and $y$ ( $x \neq y$ ) and set $a_x = \left\lceil \frac{a_x}{a_y} \right\rceil$ (ceiling function).
Your goal is to make array $a$ consist of $n - 1$ ones and $1$ two in no more than $n + 5$ steps. Note that you don't have to minimize the number of steps.
输入格式
The first line contains a single integer $t$ ( $1 \le t \le 1000$ ) — the number of test cases.
The first and only line of each test case contains the single integer $n$ ( $3 \le n \le 2 \cdot 10^5$ ) — the length of array $a$ .
It's guaranteed that the sum of $n$ over test cases doesn't exceed $2 \cdot 10^5$ .
The first and only line of each test case contains the single integer $n$ ( $3 \le n \le 2 \cdot 10^5$ ) — the length of array $a$ .
It's guaranteed that the sum of $n$ over test cases doesn't exceed $2 \cdot 10^5$ .
输出格式
For each test case, print the sequence of operations that will make $a$ as $n - 1$ ones and $1$ two in the following format: firstly, print one integer $m$ ( $m \le n + 5$ ) — the number of operations; next print $m$ pairs of integers $x$ and $y$ ( $1 \le x, y \le n$ ; $x \neq y$ ) ( $x$ may be greater or less than $y$ ) — the indices of the corresponding operation.
It can be proven that for the given constraints it's always possible to find a correct sequence of operations.
It can be proven that for the given constraints it's always possible to find a correct sequence of operations.
输入输出样例
输入 #1
2 3 4
输出 #1
2 3 2 3 2 3 3 4 4 2 4 2
In the first test case, you have array $a = [1, 2, 3]$ . For example, you can do the following:
1. choose $3$ , $2$ : $a_3 = \left\lceil \frac{a_3}{a_2} \right\rceil = 2$ and array $a = [1, 2, 2]$ ;
2. choose $3$ , $2$ : $a_3 = \left\lceil \frac{2}{2} \right\rceil = 1$ and array $a = [1, 2, 1]$ .
You've got array with $2$ ones and $1$ two in $2$ steps.In the second test case, $a = [1, 2, 3, 4]$ . For example, you can do the following:
1. choose $3$ , $4$ : $a_3 = \left\lceil \frac{3}{4} \right\rceil = 1$ and array $a = [1, 2, 1, 4]$ ;
2. choose $4$ , $2$ : $a_4 = \left\lceil \frac{4}{2} \right\rceil = 2$ and array $a = [1, 2, 1, 2]$ ;
3. choose $4$ , $2$ : $a_4 = \left\lceil \frac{2}{2} \right\rceil = 1$ and array $a = [1, 2, 1, 1]$ .
1. choose $3$ , $2$ : $a_3 = \left\lceil \frac{a_3}{a_2} \right\rceil = 2$ and array $a = [1, 2, 2]$ ;
2. choose $3$ , $2$ : $a_3 = \left\lceil \frac{2}{2} \right\rceil = 1$ and array $a = [1, 2, 1]$ .
You've got array with $2$ ones and $1$ two in $2$ steps.In the second test case, $a = [1, 2, 3, 4]$ . For example, you can do the following:
1. choose $3$ , $4$ : $a_3 = \left\lceil \frac{3}{4} \right\rceil = 1$ and array $a = [1, 2, 1, 4]$ ;
2. choose $4$ , $2$ : $a_4 = \left\lceil \frac{4}{2} \right\rceil = 2$ and array $a = [1, 2, 1, 2]$ ;
3. choose $4$ , $2$ : $a_4 = \left\lceil \frac{2}{2} \right\rceil = 1$ and array $a = [1, 2, 1, 1]$ .
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评测结果：Accepted