Question

7.4. Турнир лучников проводился по следующим правилам. С каждого участника собрали одинаковый взнос. Организато- ры турнира забрали 1/3 от всех поступивших денег, а остав- шиеся деньги пошли в призовой фонд турнира. Робин Гуд, победивший в турнире, получил больше каждого из осталь- ных участников — 1/6 от призового фонда, однако оказался в убытке. Какое количество лучников могло участвовать в турнире? Приведите все возможные варианты и докажите, что других нет.

Answer 1

Пусть участвовали n лучников, каждый из которых внёс s.

Призовой фонд окажется равным 2/3 * ns, выигрыш Робина Гуда составит 1/6 * 2/3 * ns = ns/9, и это оказалось меньше размера взноса s.

ns/9 < s

n/9 < 1

n < 9

С другой стороны, n - 1 проигравших участников суммарно получили 2ns/3 * (1 - 1/6). Следовательно, среди проигравших участников по принципу Дирихле обязательно найдётся тот, кто получил не меньше 2ns/3 * (1 - 1/6)/(n - 1), и это должно быть меньше доли, полученной Робин Гудом 2ns/3 * 1/6:

(1 - 1/6)/(n - 1) < 1/6

n - 1 > 5

n > 6

С учётом неравенства получаем 2 варианта:

n = 7, 8.

Answer 2

To find the number of archers that could have participated in the tournament, we need to solve the problem using the given information. Let's break it down step by step:

1. Each participant contributed an equal amount of money as an entry fee. 2. The tournament organizers took 1/3 of the total money collected as their share. 3. The remaining money was used as the prize fund for the tournament. 4. The winner, Robin Hood, received more than any other participant, specifically 1/6 of the prize fund. 5. However, Robin Hood ended up with a loss.

Let's assume the total amount of money collected as the entry fee is represented by 'x'. According to the given information:

1. The organizers took 1/3 of the total money collected, which is (1/3)x. 2. The remaining money, which went into the prize fund, is (2/3)x. 3. Robin Hood received 1/6 of the prize fund, which is (1/6)(2/3)x = (1/9)x.

Now, we know that Robin Hood ended up with a loss, which means the amount he received was less than the entry fee he paid. Therefore, we can set up the following inequality:

(1/9)x < x

Simplifying the inequality:

1/9 < 1

Since the inequality is true, we can conclude that there is at least one possible value for 'x' that satisfies the given conditions. This means there is at least one possible number of archers that could have participated in the tournament.

To find all the possible values for 'x' and the corresponding number of archers, we need to consider the factors of 'x' that satisfy the inequality. Let's list them:

1. x = 9 - The organizers took 1/3 of 9, which is 3. - The remaining prize fund is 6. - Robin Hood received 1/6 of the prize fund, which is 1. - In this case, there were 9 archers participating in the tournament.

2. x = 18 - The organizers took 1/3 of 18, which is 6. - The remaining prize fund is 12. - Robin Hood received 1/6 of the prize fund, which is 2. - In this case, there were 18 archers participating in the tournament.

Therefore, the possible number of archers that could have participated in the tournament are 9 and 18. These are the only two valid solutions that satisfy the given conditions.

Please note that the above solution is derived based on the given information and mathematical reasoning.