Show number of ways to choose 4 of n lines to form quadrilateral into which a ....

[Jashansandhu20]

1. Out of n straight lines whose lengths are 1, 2, 3, ..., n inches, respectively, show that the number of ways in which 4 may be chosen will form a quadrilateral in which a circle may be inscribed is:

. . . . .\(\displaystyle \dfrac{1}{48}\, \left[2n\, (n\, -\, 2)\, (2n\, -\, 5)\, -\, 3\, +\, (-1)^n\right]\)

2. Prove that if (p, q) = 1, then:

. . . . .\(\displaystyle \left[\dfrac{p}{q}\right]\, +\, \left[\dfrac{2p}{q}\right]\, +\, \left[\dfrac{3p}{q}\right]\, +\, ...\, +\, \left[\dfrac{(q\, -\, 1)\, p}{q}\right]\)

. . . . . . . .\(\displaystyle =\, \left[\dfrac{q}{p}\right]\, +\, \left[\dfrac{2q}{p}\right]\, +\, \left[\dfrac{3q}{p}\right]\, +\, ...\, +\, \left[\dfrac{(p\, -\, 1)\, q}{p}\right]\)

Here above, "[]" denotes the Greatest Integer Function.

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Please somebody solve these

 

[Deleted member 4993]

1. Out of n straight lines whose lengths are 1, 2, 3, ..., n inches, respectively, show that the number of ways in which 4 may be chosen will form a quadrilateral in which a circle may be inscribed is:

. . . . .\(\displaystyle \dfrac{1}{48}\, \left[2n\, (n\, -\, 2)\, (2n\, -\, 5)\, -\, 3\, +\, (-1)^n\right]\)

2. Prove that if (p, q) = 1, then:

. . . . .\(\displaystyle \left[\dfrac{p}{q}\right]\, +\, \left[\dfrac{2p}{q}\right]\, +\, \left[\dfrac{3p}{q}\right]\, +\, ...\, +\, \left[\dfrac{(q\, -\, 1)\, p}{q}\right]\)

. . . . . . . .\(\displaystyle =\, \left[\dfrac{q}{p}\right]\, +\, \left[\dfrac{2q}{p}\right]\, +\, \left[\dfrac{3q}{p}\right]\, +\, ...\, +\, \left[\dfrac{(p\, -\, 1)\, q}{p}\right]\)

Here above, "[]" denotes the Greatest Integer Function.

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Please somebody solve these

What does (p,q) =1 mean? Are you doing modular arithmetic?

 

[lookagain]

We DO NOT do homework.

Read the rules please.

Not only that, but I'd bet these are math competition problems.

What does (p,q) =1 mean? Are you doing modular arithmetic?

It should mean the greatest common factor of p and q equals 1.