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Triangles or quadrilaterals from the vertices of a regular polygon

This type of question, asked 3 times: 6 Apr 2024, 7 Apr 2025, 2 Apr 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    6 Apr 2024

    76
    The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

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  2. Question 2

    7 Apr 2025

    140
    Let pp be the number of all triangles that can be formed by joining the vertices of a regular polygon PP of nn sides and qq be the number of all quadrilaterals that can be formed by joining the vertices of P. If p+q=126p+q= 126, then the eccentricity of the ellipse x216+y2n=1\frac{x^{2}}{16}+\frac{y^{2}}{n}= 1 is

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  3. Question 3

    2 Apr 2026 S2

    57
    Let pnp_{n} denote the total number of triangles formed by joining the vertices of an n -side regular polygon. If pn+1−pn=66p_{n+ 1}-p_{n}= 66, then the sum of all distinct prime divisors of n is :

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