Question: Which of the following statements must be true?
a. IOC member from New York must have voted for Paris in round 2.
b. IOC member from Beijing voted for London in round 3.
Explanation:
Let there be x members in the IOC.
As a member cannot vote if his or her city is in contention, the number of voters in Round 1 (R1) = x – 4
The number of voters in Round 2 (R2) = x – 3 and
The number of voters in Round 3 (R3) = x – 2 – n
Where n is the number of voters who have voted for New York (NY) in R1 and Beijing (B) in R2.
Since x – 3 = 83, we get x – 4 = 82 and x – 2 – n = 75 or n = 9
21 members voted for B in R2. Out of these, 9 voted for NY in R1.
The remaining 12 who voted for B comprised 75% of those who voted for B in R1.
Thus 12/0.75 = 16 members voted for B in R1.
∴ Paris (P) got 82 – 16 – 30 – 12 = 24 votes in R1.
All those who voted for London (L) and P in R1 continued to vote for the same cities in subsequent rounds. Thus, 24 voters of P in R2 had voted for P in R1 too. Also from the given information, 3 voters who had voted for NY in R1 voted for Paris in R2.
Out of the remaining 5 that voted for P in R2, 4 had voted for Beijing in R1 and 1 vote came from the member who represented NY.
In R3, the difference in the votes cast for L and P was 1, i.e. L and P got 37 and 38 votes in some order.
The composition of 75 voters of R3 was as follows:
12 members who had voted for B in R1 and R2 were eligible for voting in R3.
30 and 24 members who voted for L and P in R1 continued to do so in R3.
4 voters of R3, voted for B in R1 and P in R2.
3 voters of R3, voted for NY in R1 and P in R2.
1 member represented NY and 1 represented B.
From given information, 50% of voters of B in R1 i.e. 8 voted for P in R3. So, 8 out of the 12 who voted for B in R1 and R2, voted for London in R3.
The information can be summarised as shown in the table:
It can be clearly seen, that only statement a is true.
Hence, option (a).