# The television contest

In a televised contest two teams compete by performing different tests. The winner of each test receives a fixed amount of points (the same in all tests) and the loser receives a fixed amount of points less than that of the winner.

After several tests, one team has 231 points and another, which won exactly 3 tests, has 176 points.

How many points do the winner and loser receive in each test?

#### Solution

Each time you compete in the program, the difference between both teams increases or decreases by a certain amount, depending on whether they lose or win. Since the final scores are 231 for one team and 176 for the other, it is clear that the first won more than the second, that is, more than three times, and the team with the lowest score won three times.

Note that the difference between the two teams is 231 - 176 = 55, so 55 must be a multiple of the difference between the points given to the winner, and those given to the loser. According to this, we have 4 possible differences, 1, 5, 11, and 55, which are all divisors.

If the difference between what they receive is 55, the first team should have won 4 times and the other, 3. Adding both results then, it would be the total of having won and lost 7 times, that is, 231 + 176 = 407 It would be divisible by 7, and it isn't. Therefore it cannot be 55.

In the same way, if it were 11, then the first team would have won 8 times and the second, 3. And since 407 is a multiple of 8 + 3 = 11, it could be that the sum of what they earn in both cases is 37 , because 407 = 37 * 11. Since the difference is 11, if 37 we take 11, there must be twice what you get when you lose, that is, 26. That means that when you lose they give you 13 and when you win, 24. Since 13 * 3 + 8 * 24 = 39 + 192 = 231, and 13 * 8 + 24 * 3 = 104 + 72 = 176, it is perfectly possible.

Notice that what they give to both of them together is twice what they give to the loser, plus the difference between what the winner earns and what the loser earns, so you can know.