Question

Instructions:

In a chess tournament, 64 players ranked 1 to 64 participated. Following were the rules of this knockout tournament:

1. There were six rounds in the tournament.
2. In the first round:
   • Match 1 was played between player Ranked 1 vs. player Ranked 64.
   • Match 2 was played between player Ranked 2 vs. player Ranked 63, and so on.
3. In the second round:
   • Match 1 was played between the winner of Match No. 1 of the first round vs. the winner of Match No. 32 of the first round.
   • Similarly, Match 2 was played between the winner of Match No. 2 of the first round vs. the winner of Match No. 31 of the first round.
4. Next, all rounds were played with the above-mentioned rules.

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Questions:

Question 84:
If Ranked 43 won the tournament, then which of the following players cannot be the runner-up?

Options:
(A) Player Ranked 44
(B) Player Ranked 45
(C) Player Ranked 46
(D) Player Ranked 36

Options:
(1) A only
(2) B and D
(3) A and C
(4) C only

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Question 85:
Who could be the lowest-ranked player facing the player seeded 29 in the finals?

Options:
(1) 57
(2) 59
(3) 62
(4) 63

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Question 86:
If one of the matches was between the players seeded 14 and 55, then one of the matches in the tournament can be between players seeded:

Options:
(A) 25 and 52
(B) 43 and 47
(C) 7 and 39
(D) 2 and 6

Options:
(1) B and D
(2) A and C
(3) D and B
(4) C only

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Question 87:
If there are only five such cases when a lower-ranked player beats a higher-ranked player in the tournament, then who could be the lowest-ranked player winning the tournament?

Options:
(1) 32
(2) 16
(3) 17
(4) 63

Answer 1

Question 84:

If Ranked 43 won the tournament, which of the following players cannot be the runner-up?

Rules Recap:

• Player Ranked 43 can only face opponents in later rounds who won their respective matches.
• Runner-up must lose only to the champion, Ranked 43.

Analysis:

1. Player Ranked 44:
   Ranked 44 would have faced Ranked 21 in the first round, which they would lose because higher-ranked players win unless stated otherwise. Hence, Ranked 44 cannot be the runner-up.

2. Player Ranked 45:
   Ranked 45 would face Ranked 20 in the first round and lose. Thus, they also cannot be the runner-up.

3. Player Ranked 46:
   Ranked 46 would lose to Ranked 19 in the first round. Hence, they cannot reach the finals.

4. Player Ranked 36:
   Ranked 36 could win against Ranked 29 in the first round, and under the given rules, could reach the finals.

Answer: (3) A and C

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Question 85:

Who could be the lowest-ranked player facing the player seeded 29 in the finals?

Rules Recap:

• The player facing Ranked 29 in the finals must win all their matches up to the finals.
• Lowest-ranked players must defeat progressively higher-ranked players in the rounds.

Analysis:

• Player Ranked 63 starts by defeating 2nd highest seed (Ranked 2) in the first round, then moves up the bracket, defeating progressively higher-ranked players.
• Ranked 63 can consistently win and reach the finals.

Answer: (4) 63

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Question 86:

If one of the matches was between players seeded 14 and 55, then one of the matches in the tournament can be between players seeded:

Rules Recap:

• Players 14 and 55 must have reached the same round.
• Pairings in earlier rounds follow the rules (1 vs 64, 2 vs 63, etc.).

Analysis:

1. Pair A (25 and 52):
   This pairing could occur in Round 2 as both players win their first matches.

2. Pair B (43 and 47):
   These two players can also face each other in a later round.

3. Pair C (7 and 39):
   These two players can face each other in a later round.

4. Pair D (2 and 6):
   This pairing is impossible because they are in separate brackets and cannot meet.

Answer: (3) D and B

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Question 87:

If there are only five cases where a lower-ranked player beats a higher-ranked player, then who could be the lowest-ranked player winning the tournament?

Rules Recap:

• Only five upsets are allowed, meaning lower-ranked players can beat higher-ranked players five times.
• The winner must have participated in six matches to win the tournament.

Analysis:

1. Ranked 32 can win the tournament by defeating only five higher-ranked players across six matches.
2. Ranked 16 and 17 would require more upsets than allowed to win.
3. Ranked 63 cannot win because it requires more than five upsets to reach the finals.

Answer: (1) 32