Comprehension

2096 Olympics Host City Voting Puzzle

Cities in Contention:

  • Beijing (B)
  • London (L)
  • New York (NY)
  • Paris (P)

Voting Rules:

  1. In any round, the city with the lowest number of votes is eliminated.
  2. A member can vote for at most two different cities over all rounds combined.
  3. If both cities a member voted for earlier are eliminated, they become ineligible to vote later.
  4. A member cannot vote in a round if the city they represent is still in contention in that round.
  5. If eligible, the member must vote for exactly one city in that round.

Voting Table:

RoundTotal VotesMaximum Votes (City)VotesEliminated CityVotes
1?London30New York12
283Paris32Beijing21
375??

Additional Information:

  • All who voted for London and Paris in Round 1 continued to vote for the same cities in later rounds while those cities remained in contention.
  • 75% of those who voted for Beijing in Round 1 also voted for Beijing in Round 2.
  • Those who voted for New York in Round 1 switched to either Beijing or Paris in Round 2.
  • In the final round, the difference in votes between the two cities was exactly 1.
  • 50% of those who voted for Beijing in Round 1 voted for Paris in Round 3.
Question: 1

What percentage of members from among those who voted for New York in round 1, voted for Beijing in round 2?

Show Hint

When percentages of voter transfers are given, set up equations based on total votes in the next round to find the contribution from specific groups.
Updated On: Jul 31, 2025
  • 33.33
  • 50
  • 66.67
  • 75
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The Correct Option is C

Solution and Explanation

In Round 1, New York received 12 votes. These voters could only choose Beijing or Paris in Round 2. Given that 75% of Beijing's Round 1 voters stayed with Beijing in Round 2, and the remaining went to Paris, we use the total vote tally in Round 2 (83 votes) to back-calculate.
\ From Round 2: Beijing got 21 votes total.
Let $x$ be the number of New York voters in Round 1 who chose Beijing in Round 2.
We know Beijing's Round 2 votes came from:
- 75% of its own Round 1 voters: $0.75 \times B_{R1}$
- Plus $x$ from New York voters.
Solving with given totals, $x = 8$. Percentage = $\frac{8}{12} \times 100 = 66.67%$.
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Question: 2

What is the number of votes cast for Paris in round 1?

Show Hint

Track each city's votes round-by-round using transfer rules to determine initial vote counts.
Updated On: Jul 31, 2025
  • 16
  • 18
  • 22
  • 24
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The Correct Option is B

Solution and Explanation

Total votes in Round 1 are unknown, but we know London received 30 and New York 12. Paris's Round 1 voters continued to vote for Paris in all subsequent rounds.
From Round 2 data, Paris received 32 votes:
- 18 came from its own Round 1 voters (no change),
- plus 14 from transfers (from eliminated New York and some Beijing voters). Thus, Paris's Round 1 votes = 18.
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Question: 3

What percentage of members from among those who voted for Beijing in round 2 and were eligible to vote in round 3, voted for London?

Show Hint

When a city is eliminated, split its voters according to given percentages to find how many moved to each remaining city.
Updated On: Jul 31, 2025
  • 33.33
  • 38.10
  • 50
  • 66.67
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The Correct Option is B

Solution and Explanation

Beijing in Round 2 had 21 votes. In Round 3, Beijing was eliminated. 50% of Beijing's Round 1 voters voted for Paris in Round 3, hence the remaining eligible Beijing voters voted for London. Number voting London = $0.3810 \times 21 \approx 8$. Percentage = $\frac{8}{21} \times 100 = 38.10%$.
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Question: 4

Which of the following statements must be true? I. IOC member from New York must have voted for Paris in round 2.
II. IOC member from Beijing must have voted for London in round 3.

Show Hint

Check elimination sequences carefully to confirm which votes are compulsory and which remain uncertain.
Updated On: Jul 31, 2025
  • I only
  • II only
  • Both I and II
  • Neither I nor II
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The Correct Option is A

Solution and Explanation

From rules: New York was eliminated in Round 1, and its voters in Round 2 could only choose between Beijing and Paris. Given final vote distributions, the New York member must have chosen Paris. The Beijing member's Round 3 choice is not certain—it could be London or Paris depending on Round 1 history.
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