Answer :
In this problem, three people—Achilles, Agatha, and Aleksanteri—are running around a circular track of length 'D' metres. We need to figure out which value for 'R', the number of rounds raced, is impossible given the following conditions:
- Achilles beats Agatha by 800 metres.
- Achilles beats Aleksanteri by 1200 metres.
- Agatha beats Aleksanteri by 600 metres.
Since we're dealing with a circular track, understand the following:
- When Achilles finishes the race of [tex]R \times D[/tex] metres, Agatha is 800 metres behind. This means Agatha has run [tex]R \times D - 800[/tex] metres.
- Aleksanteri is 1200 metres behind Achilles, so Aleksanteri has run [tex]R \times D - 1200[/tex] metres.
- Since Agatha beats Aleksanteri by 600 metres, when Agatha completes [tex]R \times D - 800[/tex] metres, Aleksanteri has run [tex]R \times D - 1400[/tex] metres.
Let's now consider the strategies for which R could be impossible.
- Track Length Relations: For one full round, the meters run must be [tex]R[/tex] times the track length for the finish line. If the total distance run by each is not equivalent to an integer multiple of the track, that's a key issue.
Evaluating the differences:
- The 800 and 1200 meters suggest modulo [tex]D[/tex].
- Thus, [tex]R \times D \equiv 0 \pmod{D}[/tex]
If there's a mismatched value in R such that there isn’t a consistent integer result for a single person's run—specifically the missed beats—they likely don't satisfy equal integer multiples for certain given permissible D-lengths.
Hence, trial and evaluation of all options show:
- Option E: 15 is particularly inconsistent since commonly normalized intervals (which are likely the possible actual laps) aren't divisible to suit transitions using 800 and 600 together precisely.
Evaluating:
- Checking common multiples:
- 800, 600 fit multiples more precisely such as 2400, but mismatches arise inconclusively at 15,
- And gaps arise to test convincingly again, even overwriting backverifies respectably, edges don’t strongly reconcile either remaining meters between laps without already missed listed disqualifiers.
Therefore, R = 15 is not possible with integer length 'D' and conditions satisfied.
The correct option is E) 15.
