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Problem 747


Problem 747


Triangular Pizza

Mamma Triangolo baked a triangular pizza. She wants to cut the pizza into n pieces. She first chooses a point P in the interior (not boundary) of the triangle pizza, and then performs n cuts, which all start from P and extend straight to the boundary of the pizza so that the n pieces are all triangles and all have the same area.

Let ψ(n) be the number of different ways for Mamma Triangolo to cut the pizza, subject to the constraints.
For example, ψ(3)=7.

Also ψ(6)=34, and ψ(10)=90.

Let Ψ(m)=n=3mψ(n). You are given Ψ(10)=345 and Ψ(1000)=172166601.

Find Ψ(108). Give your answer modulo 1 000 000 007.


三角形披萨

特莱格伦夫人打算把一块三角形的披萨分成n片。她先在三角形披萨的内部(而非边缘)选择一个点P,然后从点P出发切n刀至披萨边缘,将披萨分成n个面积相同的小三角形。

ψ(n)为满足特莱格伦夫人上述要求的披萨切法。
例如,ψ(3)=7

此外,ψ(6)=34ψ(10)=90

Ψ(m)=n=3mψ(n)。已知Ψ(10)=345Ψ(1000)=172166601

Ψ(108),并将你的答案对1 000 000 007取余。


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