Task 6 - 121046
Two caravans set out from oasis A at the same time and marched on the same route via B and C to D.

The first caravan marched for three days in a row and then took a rest day, while the second caravan marched for two days in a row and then took two rest days.

Both caravans set out on marching days at the same time and were each the same number of hours on the move. They reached the destinations B, C, D at the end of these hours of a marching day. During their marching days, each caravan maintained a constant speed.

The first caravan needed twice as many days for the journey from A to C, including rest days, and three times as many days for the journey from A to B, including rest days.

Both caravans arrived in B at the end of a marching day.

Determine whether the caravans also arrived in D at the same time! If not, determine which of the two caravans arrived in D first!
}

Let \(n=4 q+r\) with \(n, q, r \in \mathbb{Z}_{\geq 0}\) and \(0 \leq r<4\) be the number of days both caravans are on the way from A to B. Since the first caravan does not arrive in C after a number of days divisible by 4 (which would be one of their rest days), \(n\) cannot be even (since it reaches C after \(2 n\) days). Therefore, \(r\) must be either 1 or 3.

The latter cannot be the case, however, because the second caravan takes a rest day on all such representable days, and thus could not have arrived in B simultaneously with the first caravan. Therefore, there exists a non-negative integer \(q\) such that both caravans reach B after exactly \(4 q+1\) days.

Thus, the first caravan arrives at the destination after \(12 q+3\) days and has marched for \(9 q+3\) days during this time; of these, it marched for \(3 q+1\) days from A to B, so the total distance from A to D is exactly three times as long as the segment from A to B.

The second caravan covered this segment with \(2 q+1\) marching days, so it needed a total of \(6 q+3\) marching days for the entire journey from A to D. The \((6 q+3)\)-rd marching day is exactly the \((2 \cdot (6 q+2) + 1) = 12 q + 5\)-th day after departure from A, so this second caravan arrives in D exactly two days after the first one.