# Kickstart Round A 2017 做题记录

### Problem

Mr. Panda has recently fallen in love with a new game called Square Off, in which players compete to find as many different squares as possible on an evenly spaced rectangular grid of dots. To find a square, a player must identify four dots that form the vertices of a square. Each side of the square must have the same length, of course, but it does not matter what that length is, and the square does not necessarily need to be aligned with the axes of the grid. The player earns one point for every different square found in this way. Two squares are different if and only if their sets of four dots are different.

Mr. Panda has just been given a grid with R rows and C columns of dots. How many different squares can he find in this grid? Since the number might be very large, please output the answer modulo 109 + 7 (1000000007).

### Input

The first line of the input gives the number of test cases, T. T lines follow. Each line has two integers R and C: the number of dots in each row and column of the grid, respectively.

### Output

For each test case, output one line containing `Case #x: y`, where `x` is the test case number (starting from 1) and `y` is the number of different squares can be found in the grid.

1 ≤ T ≤ 100.

2 ≤ R ≤ 1000.
2 ≤ C ≤ 1000.

2 ≤ R ≤ 109.
2 ≤ C ≤ 109.

### Sample

Input Output

The pictures below illustrate the grids from the three sample cases and a valid square in the third sample case.

$\sum _{i=2} ^{min(n,m)} (n-i+1)*(m-i+1)*(i-1)=\sum _{i=2} ^{min(n,m)} n*m(i-1)-(n+m)*(i-1)^{2}+(i-1)^{3}$公式大概这个样子吧。。。

### Problem

Alice likes reading and buys a lot of books. She stores her books in two boxes; each box is labeled with a pattern that matches the titles of all of the books stored in that box. A pattern consists of only uppercase/lowercase English alphabet letters and stars (`*`). A star can match between zero and four letters. For example, books with the titles `GoneGirl` and `GoneTomorrow` can be put in a box with the pattern `Gone**`, but books with the titles `TheGoneGirl`, and `GoneWithTheWind` cannot.

Alice is wondering whether there is any book that could be stored in either of the boxes. That is, she wonders if there is a title that matches both boxes’ patterns.

### Input

The first line of the input gives the number of test cases, T. T test cases follow. Each consists of two lines; each line has one string in which each character is either an uppercase/lowercase English letter or `*`.

### Output

For each test case, output one line containing `Case #x: y`, where `x` is the test case number (starting from 1) and `y` is `TRUE` if there is a string that matches both patterns, or `FALSE` if not.

### Limits

1 ≤ T ≤ 50.

#### Small dataset

1 ≤ the length of each pattern ≤ 200.
Each pattern contains at most 5 stars.

#### Large dataset

1 ≤ the length of each pattern ≤ 2000.

### Sample

Input Output

In sample case #1, the title `It` matches both patterns. Note that it is possible for a `*` to match zero characters.

In sample case #2, the title `Shakespeare` matches both patterns.

In sample case #3, there is no title that matches both patterns. `Shakespeare`, for example, does not work because the `*` at the start of the `*peare` pattern cannot match six letters.

C不大会，待补。。。

# Kickstart Practice Round 2017做题记录

Problem A. Country Leader

### Problem

The Constitution of a certain country states that the leader is the person with the name containing the greatest number of different alphabet letters. (The country uses the uppercase English alphabet from A through Z.) For example, the name `GOOGLE` has four different alphabet letters: E, G, L, and O. The name `APAC CODE JAM` has eight different letters. If the country only consists of these 2 persons, `APAC CODE JAM` would be the leader.

If there is a tie, the person whose name comes earliest in alphabetical order is the leader.

Given a list of names of the citizens of the country, can you determine who the leader is?

### Input

The first line of the input gives the number of test cases, T. T test cases follow. Each test case starts with a line with an interger N, the number of people in the country. Then N lines follow. The i-th line represents the name of the i-th person. Each name contains at most 20 characters and contains at least one alphabet letter.

### Output

For each test case, output one line containing `Case #x: y`, where `x` is the test case number (starting from 1) and y is the name of the leader.

### Limits

1 ≤ T ≤ 100.
1 ≤ N ≤ 100.

#### Small dataset

Each name consists of at most 20 characters and only consists of the uppercase English letters `A` through `Z`.

#### Large dataset

Each name consists of at most 20 characters and only consists of the uppercase English letters `A` through `Z` and ‘ ‘(space).
All names start and end with alphabet letters.

### Sample

Input Output

In sample case #1, `JOHNSON` contains 5 different alphabet letters(‘H’, ‘J’, ‘N’, ‘O’, ‘S’), so he is the leader.

Sample case #2 would only appear in Large data set. The name `DEF` contains 3 different alphabet letters, the name `A AB C` also contains 3 different alphabet letters. `A AB C` comes alphabetically earlier so he is the leader.

Problem B. Vote

## Problem

A and B are the only two candidates competing in a certain election. We know from polls that exactly N voters support A, and exactly M voters support B. We also know that N is greater than M, so A will win.

Voters will show up at the polling place one at a time, in an order chosen uniformly at random from all possible (N + M)! orders. After each voter casts their vote, the polling place worker will update the results and note which candidate (if any) is winning so far. (If the votes are tied, neither candidate is considered to be winning.)

What is the probability that A stays in the lead the entire time — that is, that A will always be winning after every vote?

### Input

The input starts with one line containing one integer T, which is the number of test cases. Each test case consists of one line with two integers N and M: the numbers of voters supporting A and B, respectively.

### Output

For each test case, output one line containing `Case #x: y`, where `x` is the test case number (starting from 1) and `y` is the probability that A will always be winning after every vote.

`y` will be considered correct if `y` is within an absolute or relative error of 10-6 of the correct answer. See the FAQ for an explanation of what that means, and what formats of real numbers we accept.

### Limits

1 ≤ T ≤ 100.

0 ≤ M < N ≤ 10.

#### Large dataset

0 ≤ M < N ≤ 2000.

### Sample

Input Output

In sample case #1, there are 3 voters. Two of them support A — we will call them A1 and A2 — and one of them supports B. They can come to vote in six possible orders: A1 A2 B, A2 A1 B, A1 B A2, A2 B A1, B A1 A2, B A2 A1. Only the first two of those orders guarantee that Candidate A is winning after every vote. (For example, if the order is A1 B A2, then Candidate A is winning after the first vote but tied after the second vote.) So the answer is 2/6 = 0.333333…

In sample case #2, there is only 1 voter, and that voter supports A. There is only one possible order of arrival, and A will be winning after the one and only vote.

Problem C. Sherlock and Parentheses

### Problem

Sherlock and Watson have recently enrolled in a computer programming course. Today, the tutor taught them about the balanced parentheses problem. A string `S` consisting only of characters `(` and/or `)` is balanced if:

• It is the empty string, or:
• It has the form `(`S`)`, where S is a balanced string, or:
• It has the form S1S2, where S1 is a balanced string and S2 is a balanced string.

Sherlock coded up the solution very quickly and started bragging about how good he is, so Watson gave him a problem to test his knowledge. He asked Sherlock to generate a string S of L + R characters, in which there are a total of L left parentheses `(` and a total of R right parentheses `)`. Moreover, the string must have as many different balanced non-empty substrings as possible. (Two substrings are considered different as long as they start or end at different indexes of the string, even if their content happens to be the same). Note that S itself does not have to be balanced.

Sherlock is sure that once he knows the maximum possible number of balanced non-empty substrings, he will be able to solve the problem. Can you help him find that maximum number?

### Input

The first line of the input gives the number of test cases, T. T test cases follow. Each test case consists of one line with two integers: L and R.

### Output

For each test case, output one line containing `Case #x: y`, where `x` is the test case number (starting from 1) and `y` is the answer, as described above.

1 ≤ T ≤ 100.

0 ≤ L ≤ 20.
0 ≤ R ≤ 20.
1 ≤ L + R ≤ 20.

0 ≤ L ≤ 105.
0 ≤ R ≤ 105.
1 ≤ L + R ≤ 105.

### Sample

Input Output

In Case 1, the only possible string is `(`. There are no balanced non-empty substrings.
In Case 2, the optimal string is `()`. There is only one balanced non-empty substring: the entire string itself.
In Case 3, both strings `()()(` and `(()()` give the same optimal answer.
For the case `()()(`, for example, the three balanced substrings are `()` from indexes 1 to 2, `()` from indexes 3 to 4, and `()()` from indexes 1 to 4.