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Problem 247


Problem 247


Squares under a hyperbola

Consider the region constrained by 1 ≤ x and 0 ≤ y ≤ 1/x.

Let S1 be the largest square that can fit under the curve.
Let S2 be the largest square that fits in the remaining area, and so on.
Let the index of Sn be the pair (left, below) indicating the number of squares to the left of Sn and the number of squares below Sn.

The diagram shows some such squares labelled by number.
S2 has one square to its left and none below, so the index of S2 is (1,0).
It can be seen that the index of S32 is (1,1) as is the index of S50.
50 is the largest n for which the index of Sn is (1,1).

What is the largest n for which the index of Sn is (3,3)?


双曲线下的正方形

考虑由1 ≤ x和0 ≤ y ≤ 1/x所表示的区域。

记S1是能够放进曲线下区域的最大正方形。
记S2是能够放进剩余区域的最大正方形,并依此类推。
Sn索引是一个数对(left, below),分别表示Sn左侧的正方形数目和Sn下放的正方形数目。

上图展示了一些这样的正方形,并标上了相应的数字顺序。
S2的左侧有一个正方形,下方没有正方形,因此S2的索引为(1,0)。
可以看出S32的索引和S50的索引同为(1,1)。
使得Sn索引为(1,1)的数n最大是50。

使得Sn索引为(3,3)的数n最大是多少?