Challenging Mathematical Problems with Elementary Solutions by A.M. Yaglom & I.M. Yaglom

Author:A.M. Yaglom & I.M. Yaglom
Language: eng
Format: epub, pdf
Publisher: Dover Publications

Fig. 51

We thus have

But

by formula (1) on page 107, with n replaced by n + 1.

Consequently,

and similarly

Adding these equations, we obtain

Since6

we end up with

Consequently,

Setting n = 3, 4, 5,.... , in this formula, we obtain:

f3=l, f4 = 4, f5=11, f6 = 25, f7 = 50, f8 = 82, . . .

From this result it follows in particular that if no three diagonals of a convex polygon are concurrent, then the number of pieces into which the polygon is divided by its diagonals depends only on the number of vertices and not on the shape of the polygon.

Second solution. The diagonals of an n-gon divide it into smaller polygons. We denote by r3 the number of triangles among these polygons, by r4 the number of quadrilaterals, by r5 the number of pentagons, etc., and finally by rm the number of m-gons, where m is the greatest number of sides of any of the polygons formed by the diagonals of the n-gon. We have to evaluate the sum

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