Problem Statement

Two characters x and y are called similar characters if and only if one of the following conditions is satisfied:

• x and y are the same character.
• One of x and y is 1 and the other is l.
• One of x and y is 0 and the other is o.

Two strings S and T, each of length N, are called similar strings if and only if:

• for all i\ (1\leq i\leq N), the i-th character of S and the i-th character of T are similar characters.

Given two length-N strings S and T consisting of lowercase English letters and digits, determine if S and T are similar strings.

Constraints

• N is an integer between 1 and 100.
• Each of S and T is a string of length N consisting of lowercase English letters and digits.

Sample Input 1

3
l0w
1ow

Sample Output 1

Yes

The 1-st character of S is l, and the 1-st character of T is 1. These are similar characters.

The 2-nd character of S is 0, and the 2-nd character of T is o. These are similar characters.

The 3-rd character of S is w, and the 3-rd character of T is w. These are similar characters.

Thus, S and T are similar strings.

Sample Input 2

3
abc
arc

Sample Output 2

No

The 2-nd character of S is b, and the 2-nd character of T is r. These are not similar characters.

Thus, S and T are not similar strings.

Sample Input 3

4
nok0
n0ko

Sample Output 3

Yes

Problem Statement

There is one humidifier in the AtCoder company office. The current time is 0, and the humidifier has no water inside.

You will add water to this humidifier N times. The i-th addition of water (1 \leq i \leq N) takes place at time T_i, and you add V_i liters of water. It is guaranteed that T_i < T_{i+1} for all 1 \leq i \leq N-1.

However, the humidifier has a leak, and as long as there is water inside, the amount of water decreases by 1 liter per unit time.

Find the amount of water remaining in the humidifier immediately after you finish adding water at time T_N.

Constraints

• 1 \leq N \leq 100
• 1 \leq T_i \leq 100 (1 \leq i \leq N)
• 1 \leq V_i \leq 100 (1 \leq i \leq N)
• T_i < T_{i+1} (1 \leq i \leq N-1)
• All input values are integers.

Sample Input 1

4
1 3
3 1
4 4
7 1

Sample Output 1

3

At each point in time, water is added as follows:

• Time 1: Before adding, the humidifier has 0 liters. After adding 3 liters, it has 3 liters.
• Time 3: Before adding, it has 1 liter. After adding 1 liter, it has 2 liters total.
• Time 4: Before adding, it has 1 liter. After adding 4 liters, it has 5 liters total.
• Time 7: Before adding, it has 2 liters. After adding 1 liter, it has 3 liters total.

After finishing the addition at time 7, the humidifier contains 3 liters. Thus, the answer is 3.

Sample Input 2

3
1 8
10 11
21 5

Sample Output 2

5

Sample Input 3

10
2 1
22 10
26 17
29 2
45 20
47 32
72 12
75 1
81 31
97 7

Sample Output 3

57

Problem Statement

The AtCoder amusement park has an attraction that can accommodate K people. Now, there are N groups lined up in the queue for this attraction.

The i-th group from the front (1\leq i\leq N) consists of A_i people. For all i (1\leq i\leq N), it holds that A_i \leq K.

Takahashi, as a staff member of this attraction, will guide the groups in the queue according to the following procedure.

Initially, no one has been guided to the attraction, and there are K empty seats.

1. If there are no groups in the queue, start the attraction and end the guidance.
2. Compare the number of empty seats in the attraction with the number of people in the group at the front of the queue, and do one of the following:
   • If the number of empty seats is less than the number of people in the group at the front, start the attraction. Then, the number of empty seats becomes K again.
   • Otherwise, guide the entire group at the front of the queue to the attraction. The front group is removed from the queue, and the number of empty seats decreases by the number of people in the group.
3. Go back to step 1.

Here, no additional groups will line up after the guidance has started. Under these conditions, it can be shown that this procedure will end in a finite number of steps.

Determine how many times the attraction will be started throughout the guidance.

Constraints

• 1\leq N\leq 100
• 1\leq K\leq 100
• 1\leq A_i\leq K\ (1\leq i\leq N)
• All input values are integers.

Sample Input 1

7 6
2 5 1 4 1 2 3

Sample Output 1

4

Initially, the seven groups are lined up as follows:

Part of Takahashi's guidance is shown in the following figure:

• Initially, the group at the front has 2 people, and there are 6 empty seats. Thus, he guides the front group to the attraction, leaving 4 empty seats.
• Next, the group at the front has 5 people, which is more than the 4 empty seats, so the attraction is started.
• After the attraction is started, there are 6 empty seats again, so the front group is guided to the attraction, leaving 1 empty seat.
• Next, the group at the front has 1 person, so they are guided to the attraction, leaving 0 empty seats.

In total, he starts the attraction four times before the guidance is completed. Therefore, print 4.

Sample Input 2

7 10
1 10 1 10 1 10 1

Sample Output 2

7

Sample Input 3

15 100
73 8 55 26 97 48 37 47 35 55 5 17 62 2 60

Sample Output 3

8

Problem Statement

The digit sum of a positive integer n is defined as the sum of the digits when n is represented in decimal. For example, the digit sum of 2026 is 2+0+2+6=10.

Find the number of positive integers not exceeding N whose digit sum is K.

Constraints

• 1\leq N,K \leq 10^5
• All input values are integers.

Sample Input 1

30 4

Sample Output 1

3

Among the positive integers not exceeding 30, there are three integers whose digit sum is 4: 4,13,22.

Sample Input 2

2026 10

Sample Output 2

121

Sample Input 3

99999 45

Sample Output 3

1

Problem Statement

You are given a sequence A=(A_1,A_2,\dots,A_{3N}) of length 3N where each of 1,2,\dots, and N occurs exactly three times.

For i=1,2,\dots,N, let f(i) be the index of the middle occurrence of i in A. Sort 1,2,\dots,N in ascending order of f(i).

Formally, f(i) is defined as follows.

• Suppose that those j such that A_j = i are j=\alpha,\beta,\gamma\ (\alpha < \beta < \gamma). Then, f(i) = \beta.

Constraints

• 1\leq N \leq 10^5
• 1 \leq A_j \leq N
• i occurs in A exactly three times, for each i=1,2,\dots,N.
• All input values are integers.

Sample Input 1

3
1 1 3 2 3 2 2 3 1

Sample Output 1

1 3 2

• 1 occurs in A at A_1,A_2,A_9, so f(1) = 2.
• 2 occurs in A at A_4,A_6,A_7, so f(2) = 6.
• 3 occurs in A at A_3,A_5,A_8, so f(3) = 5.

Thus, f(1) < f(3) < f(2), so 1,3, and 2 should be printed in this order.

Sample Input 2

1
1 1 1

Sample Output 2

1

Sample Input 3

4
2 3 4 3 4 1 3 1 1 4 2 2

Sample Output 3

3 4 1 2

Problem Statement

There is a grid with H horizontal rows and W vertical columns. For two integers i and j such that 1 \leq i \leq H and 1 \leq j \leq W, the square at the i-th row from the top and j-th column from the left (which we denote by (i, j)) has an integer A_{i, j} written on it.

Takahashi is currently at (1,1). From now on, he repeats moving to an adjacent square to the right of or below his current square until he reaches (H, W). When he makes a move, he is not allowed to go outside the grid.

Takahashi will be happy if the integers written on the squares he visits (including initial (1, 1) and final (H, W)) are distinct. Find the number of his possible paths that make him happy.

Constraints

• 2 \leq H, W \leq 10
• 1 \leq A_{i, j} \leq 10^9
• All values in the input are integers.

Sample Input 1

3 3
3 2 2
2 1 3
1 5 4

Sample Output 1

3

There are six possible paths:

• (1, 1) \rightarrow (1, 2) \rightarrow (1, 3) \rightarrow (2, 3) \rightarrow (3, 3): the integers written on the squares he visits are 3, 2, 2, 3, 4, so he will not be happy.
• (1, 1) \rightarrow (1, 2) \rightarrow (2, 2) \rightarrow (2, 3) \rightarrow (3, 3): the integers written on the squares he visits are 3, 2, 1, 3, 4, so he will not be happy.
• (1, 1) \rightarrow (1, 2) \rightarrow (2, 2) \rightarrow (3, 2) \rightarrow (3, 3): the integers written on the squares he visits are 3, 2, 1, 5, 4, so he will be happy.
• (1, 1) \rightarrow (2, 1) \rightarrow (2, 2) \rightarrow (2, 3) \rightarrow (3, 3): the integers written on the squares he visits are 3, 2, 1, 3, 4, so he will not be happy.
• (1, 1) \rightarrow (2, 1) \rightarrow (2, 2) \rightarrow (3, 2) \rightarrow (3, 3): the integers written on the squares he visits are 3, 2, 1, 5, 4, so he will be happy.
• (1, 1) \rightarrow (2, 1) \rightarrow (3, 1) \rightarrow (3, 2) \rightarrow (3, 3): the integers written on the squares he visits are 3, 2, 1, 5, 4, so he will be happy.

Thus, the third, fifth, and sixth paths described above make him happy.

Sample Input 2

10 10
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100

Sample Output 2

48620

In this example, every possible path makes him happy.

Problem Statement

Takahashi and Aoki are competing in an election.
There are N electoral districts. The i-th district has X_i + Y_i voters, of which X_i are for Takahashi and Y_i are for Aoki. (X_i + Y_i is always an odd number.)
In each district, the majority party wins all Z_i seats in that district. Then, whoever wins the majority of seats in the N districts as a whole wins the election. (\displaystyle \sum_{i=1}^N Z_i is odd.)
At least how many voters must switch from Aoki to Takahashi for Takahashi to win the election?

Constraints

• 1 \leq N \leq 100
• 0 \leq X_i, Y_i \leq 10^9
• X_i + Y_i is odd.
• 1 \leq Z_i
• \displaystyle \sum_{i=1}^N Z_i \leq 10^5
• \displaystyle \sum_{i=1}^N Z_i is odd.

Sample Input 1

1
3 8 1

Sample Output 1

3

Since there is only one district, whoever wins the seat in that district wins the election.
If three voters for Aoki in the district switch to Takahashi, there will be six voters for Takahashi and five for Aoki, and Takahashi will win the seat.

Sample Input 2

2
3 6 2
1 8 5

Sample Output 2

4

Since there are more seats in the second district than in the first district, Takahashi must win a majority in the second district to win the election.
If four voters for Aoki in the second district switch sides, Takahashi will win five seats. In this case, Aoki will win two seats, so Takahashi will win the election.

Sample Input 3

3
3 4 2
1 2 3
7 2 6

Sample Output 3

0

If Takahashi will win the election even if zero voters switch sides, the answer is 0.

Sample Input 4

10
1878 2089 16
1982 1769 13
2148 1601 14
2189 2362 15
2268 2279 16
2394 2841 18
2926 2971 20
3091 2146 20
3878 4685 38
4504 4617 29

Sample Output 4

86

Problem Statement

We have a rooted tree with N vertices. The vertices are numbered 1 through N, and the root is Vertex 1.
The i-th edge connects Vertices A_i and B_i.
Vertex i has an integer X_i written on it.

You are given Q queries. For the i-th query, given a pair of integers (V_i,K_i), answer the following question.

• Question: among the integers written on the vertices in the subtree rooted at Vertex V_i, find the K_i-th largest value.

Constraints

• 2 \leq N \leq 10^5
• 0\leq X_i\leq 10^9
• 1\leq A_i,B_i\leq N
• 1\leq Q \leq 10^5
• 1\leq V_i\leq N
• 1\leq K_i\leq 20
• The given graph is a tree.
• The subtree rooted at Vertex V_i has K_i or more vertices.
• All values in input are integers.

Sample Input 1

5 2
1 2 3 4 5
1 4
2 1
2 5
3 2
1 2
2 1

Sample Output 1

4
5

The tree given in this input is shown below.

For the 1-st query, the vertices in the subtree rooted at Vertex 1 are Vertices 1, 2, 3, 4, and 5, so print the 2-nd largest value of the numbers written on these vertices, 4.
For the 2-nd query, the vertices in the subtree rooted at Vertex 2 are Vertices 2, 3, and 5, so print the 1-st largest value of the numbers written on these vertices, 5.

Sample Input 2

6 2
10 10 10 9 8 8
1 4
2 1
2 5
3 2
6 4
1 4
2 2

Sample Output 2

9
10

Sample Input 3

4 4
1 10 100 1000
1 2
2 3
3 4
1 4
2 3
3 2
4 1

Sample Output 3

1
10
100
1000

Problem Statement

You are given N integers A_1, A_2, \dots, A_N.
Find the number of triples of integers (i, j, k) that satisfy the following conditions:

• 1 \le i, j, k \le N
• A_i \times A_j = A_k

Constraints

• 1 \le N \le 1000
• \color{red}{1 \le A_i < 10^{1000}}

Sample Input 1

5
2
3
6
12
24

Sample Output 1

6

The following six triples (i, j, k) satisfy the conditions in the problem statement:

• (1, 2, 3)
• (1, 3, 4)
• (1, 4, 5)
• (2, 1, 3)
• (3, 1, 4)
• (4, 1, 5)

Sample Input 2

11
1
2
3
4
5
6
123456789123456789
123456789123456789
987654321987654321
987654321987654321
121932631356500531347203169112635269

Sample Output 2

40

Note that the values of each integer A_i can be huge.

Sample Input 3

9
4
4
4
2
2
2
1
1
1

Sample Output 3

162

Note that there may be duplicates among the values of A_i.