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Problem 94


Problem 94


Almost equilateral triangles

It is easily proved that no equilateral triangle exists with integral length sides and integral area. However, the almost equilateral triangle 5-5-6 has an area of 12 square units.

We shall define an almost equilateral triangle to be a triangle for which two sides are equal and the third differs by no more than one unit.

Find the sum of the perimeters of all almost equilateral triangles with integral side lengths and area and whose perimeters do not exceed one billion (1,000,000,000).


几乎等边的三角形

可以证明,不存在边长为整数的等边三角形其面积也是整数。但是,存在几乎等边的三角形 5-5-6,其面积恰好为12。

我们定义几乎等边的三角形是有两条边一样长,且第三边与这两边最多相差1的三角形。

对于所有边长和面积均为整数且周长不超过十亿(1,000,000,000)的三角形,求其中几乎等边的三角形的周长之和。