[Solution] Charmed by the Game solution codeforces

Charmed by the Game solution codeforces


Alice and Borys are playing tennis.

A tennis match consists of games. In each game, one of the players is serving and the other one is receiving.

Players serve in turns: after a game where Alice is serving follows a game where Borys is serving, and vice versa.

Each game ends with a victory of one of the players. If a game is won by the serving player, it’s said that this player holds serve. If a game is won by the receiving player, it’s said that this player breaks serve.

It is known that Alice won aa games and Borys won bb games during the match. It is unknown who served first and who won which games.

Find all values of kk such that exactly kk breaks could happen during the match between Alice and Borys in total.

Input Charmed by the Game solution codeforces

Each test contains multiple test cases. The first line contains the number of test cases tt (1t1031≤t≤103). Description of the test cases follows.

Each of the next tt lines describes one test case and contains two integers aa and bb (0a,b1050≤a,b≤105a+b>0a+b>0) — the number of games won by Alice and Borys, respectively.

It is guaranteed that the sum of a+ba+b over all test cases does not exceed 21052⋅105.

Output Charmed by the Game solution codeforces

For each test case print two lines.

In the first line, print a single integer mm (1ma+b+11≤m≤a+b+1) — the number of values of kk such that exactly kk breaks could happen during the match.

In the second line, print mm distinct integers k1,k2,,kmk1,k2,…,km (0k1<k2<<kma+b0≤k1<k2<…<km≤a+b) — the sought values of kk in increasing order.

Example Charmed by the Game solution codeforces
input

Copy
3
2 1
1 1
0 5
output

Copy
4
0 1 2 3
2
0 2
2
2 3
Note

In the first test case, any number of breaks between 00 and 33 could happen during the match:

  • Alice holds serve, Borys holds serve, Alice holds serve: 00 breaks;
  • Borys holds serve, Alice holds serve, Alice breaks serve: 11 break;
  • Borys breaks serve, Alice breaks serve, Alice holds serve: 22 breaks;
  • Alice breaks serve, Borys breaks serve, Alice breaks serve: 33 breaks.

In the second test case, the players could either both hold serves (00 breaks) or both break serves (22 breaks).

In the third test case, either 22 or 33 breaks could happen:

  • Borys holds serve, Borys breaks serve, Borys holds serve, Borys breaks serve, Borys holds serve: 22 breaks;
  • Borys breaks serve, Borys holds serve, Borys breaks serve, Borys holds serve, Borys breaks serve: 33 breaks.

 

 

Solution: Click here

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