Tourists & Buses et al.
1) A group of tourists is offered seats in a number of buses so that there were the same number of tourists in each bus. First the organizers tried 22 tourists in each bus, but 1 was left unseated. Then 1 bus left empty and the tourists used seats in the remaining buses. Find the number of buses and tourists if each bus has only 44 seats.
2) 2 cyclists started to ride at 8 am, one from A to B, and the other from B to A. Each rode at a constant speed along the same road and when each arrived at the terminal point, immediately turned back. They met for the first time at 11 am and each of them turned exactly once before they met for a second time. Find the time of their second meeting.
3) 1999 numbers are placed around the circumference of a circle. When any four successive numbers are added, the total is always 28. What are these 1999 numbers? Find all possibilities.
Let be the number of tourist, let be the number of buses, let be the number of seats.
Originally Posted by puellaevixlegitimae
From reading the problem if we had just one more tourist, i.e. we would be able to evenly divide them among the 22 seats so . The other equation says that if we had one bus less then the tourists can be evenly divided with seats in each both, so, .
Thus, we have two equations,
The last equation tells us that because the LHS is positive while RHS would not be.
Note that is an integer so the right hand side must be an integer.
Since by the conditions of the problem .
And the contrainst that we found we need to find all with such that is an integer.
We can throw away half the cases if we realize that the denominator and numerator have opposite parity. And so the denominator cannot be even. That happen only when is even. Thus, we need to check the cases by hand which is fast.
But I notice (without doing that) a trivial answer of for the makes the denominator 1.
Note: There might be other solutions which I did not check for.
There is a clever approach to #2 . . .
It helps if you make a sketch of their travelling.
Let be the distance between and
They started at 8 am. .At 11 am they met at .
. . It took them hours to cover a total of miles.
They continue on their way, turn at the end, and cycle towards each other again.
. . Then they meet at point .
Their total distance (since 8 am) is . (Look at your sketch.)
. . To cover miles, it takes them hours.
Therefore, their second meeting was at