Games and Tournaments Questions for CAT

A knockout (single-elimination) tournament has 16 players and needs no byes. The loser of each match is eliminated and the winner advances, until one champion remains. The rounds, in order, are Round 1, Quarter-Final, Semi-Final and Final.

In a tournament, 64 teams are participating for the title of the international champion. These 64 teams are seeded 1 to 64, where 1 is the highest seed and 64 is the lowest seed. The tournament is played in rounds, such that in each round, half the teams are eliminated and the other half qualify to the next round. Each team also has a dynamic seed, which keeps changing with each round. Dynamic seed is only calculated for teams that are still alive in a particular round and is only calculated at the beginning of a new round. In Round 1, the dynamic seed of each team is the same as their seed at the beginning of the tournament. From Round 2, it is calculated as follows:

Dynamic seed of team A = Dynamic seed of Team A in the last round or the dynamic seed of the team that Team A defeated in the last round, whichever is a higher seed.

Thus, suppose if the team seeded 2 defeats the team seeded 1 in Round 1, then at the start of the second round, the dynamic seed of the team originally seeded 2 becomes 1[since seed 1 is a higher seed than seed 2].

Round 1:

In Round 1, there are a total of 32 games played. In each game, there are two teams participating, and all 64 teams play one game each. The sum of the dynamic seeds of the two teams in any game is either 97 or 33. In a game, only one team wins. There are no ties. If in a game, a higher dynamic seeded team wins, it qualifies to the next round, and the other team is eliminated. If in a game, a lower dynamic seeded team wins, the game is replayed, and the winner of the replayed game qualifies to the next round and the original game is no longer considered valid. Round 2:

At the start of Round 2, the dynamic seeds of the remaining teams are recalculated. The 32 teams are then arranged in ascending order of their dynamic seeds. The highest dynamic seeded team plays the lowest dynamic seeded team. The second-highest dynamic seeded team plays the second-lowest dynamic seeded team and so on. Round 2 is played in the exact same way as points 3, 4, and 5 mentioned for Round 1. 16 teams qualify for Round 3. This system followed in Round 2 continues in the subsequent rounds till there are 2 finalists left. The round preceding the finals is called the semi-finals and the round preceding that is called the quarter finals.

In the final, only one game is played, and the winner of the game is declared the champion.

Based on the information answer the questions that follow.

If the team originally seeded 61 plays the team originally seeded 64 in one of the semi-finals, which two teams could have played the other semi-final?

Five teams — Team A, Team B, Team C, Team D and Team E — play a single round robin tournament in which every team plays every other team exactly once. Points are awarded as Win = 2, Draw = 1, Loss = 0. The final standings, showing each team's matches Played (P), Won (W), Drawn (D), Lost (L) and total Points (Pts), are given in the table in the image.

Five players seeded 1 to 5 (Seed 1 strongest) play a round robin, each pair once, with Win = 2, Loss = 0 and no draws. Only the final standings are known: Seed 2 finished 1st with 8 points, Seed 1 second with 6 points, Seed 4 third with 4 points, Seed 3 fourth with 2 points, and Seed 5 last with 0 points. An UPSET is a match in which a lower-seeded player (larger seed number) beats a higher-seeded player. Work out the minimum number of upsets consistent with these standings.

Three players A, B and C play a single round robin in best-of-3-sets format. By the end, A's set record is 4 sets won and 0 lost, while B's set record is 2 sets won and 3 lost. Use these records to reconstruct the results.

A football tournament was organized with six participating teams: Ahmedabad Tigers (A), Bangalore Titans (B), Chennai Superkings (C), Delhi Daredevils (D), Ernakulum Fighters (E), and Firozpur Pathans (F). Each team played every other team exactly once (5 matches per team). Scoring: win = +5, loss = -2, draw = +1 for each team.

Additional facts:

1. Total scores of B and D are equal, and score(E)+score(F)=score(C).

2. A beat B, and C beat D.

3. B beat D and F, and E beat F.

4. D beat E and F.

5. Total goals scored by E equals total goals conceded by A.

Further goal details:

i. A and C scored 1 goal versus D and E respectively.

ii. B scored 2 versus A and 2 versus E.

iii. D scored 1 versus B and 1 versus F.

iv. E scored 1 versus D, and F scored 0 versus B.

v. Goals in C vs A were (2,3) in any order, and in C vs B were (1,2) in any order.

vi. Goals in F vs C were (0,3) in any order.