[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 344
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Для чисел 1, ..., 1999, расставленных по окружности, вычисляется сумма произведений всех наборов из 10 чисел, идущих подряд. Найдите расстановку чисел, при которой полученная сумма наибольшая.
The arrangement is as follows: first all odd numbers in ascending order up to 1999, then all even numbers in descending order from 1998 to 2.
1, 3, 5, ...,1997, 1999, 1998, 1996, ...6, 4, 2 (close the circle).
MD, prove it.
Да нет, grell, просто олимпиада 1999-го года. В каждой подобные задачи встречаются.
MD, докажи.
What's there to prove, you check it out! ;)
А чё там доказывать, ты проверь! ;)
Just kidding.
The idea is this: The greatest contribution can be made by multiplication of large numbers by each other. That's why they need to be compacted.
Then act like this: put the biggest number (1999) in the middle and start to place the other big numbers as densely as possible around it.
They'll naturally alternate (one to the left, one to the right... etc.). Let's see what we've got. The result is what I wrote in my answer.
There is a circular hole in a meadow that is shaped like a square. A grasshopper jumps across the meadow. Before each jump, it chooses a peak and jumps towards it. The length of the jump is half the distance to this peak.
Can the grasshopper hit the hole?
The hole is probably small (small compared to the length of the side of the square). And the grasshopper is apparently positioned at an arbitrary point inside the square to begin with.
Вероятно, лунка маленькая (небольшого размера в сравнении с длиной стороны квадрата). А кузнечик, видимо, вначале расположен в произвольной точке внутри квадрата.
Is the hole in an arbitrary place?
// If in the centre, the problem is solved in 151 strokes anyway.
The starting point can be anything, and in this case, the solution probably comes down to being less than any given epsilon from the centre of the hole.
Ты хочешь сказать, что попадешь в лунку в центре не более чем за 151 ход, даже если она будет математической точкой? Не верю.
Начальная точка может быть любой, и в данном случае, вероятно, решение сводится к тому, чтобы оказаться на расстоянии меньше любого заданного эпсилон от центра лунки.
You didn't answer the question. Admit it, where's the hole?!
;)