how to do these problems

[golam]

G

 

[mmm4444bot]

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[stapel]

The following appears to be the text of the various exercises:

26. Let ABCD be a cyclic quadrilateral with AB = 1, BC = 2, CD = 3, and DA = 4. Find the square of the area of the quadrilateral ABCD.

27. Five men and five women stand in a circle in random order. The probability that every man stands next to at least one woman is m/n, where m and n are relatively prime integers. Find m + n.

28. Find the number of ordered triples of positive integers (a, b, c) such that abc divides (ab + 1)(bc + 1)(ca + 1).

29. Consider the sequences of six positive integers a₁, a₂, a₃, a₄, a₅, and a₆, with the properties that a₁ = 1 and if for some j > 1, a_j = m > 1, then m - 1 appears in the sequence a₁, a₂, ..., a_j-1. Such sequences include 1, 1, 2, 1, 3, 2 and 1, 2, 3, 1, 4, 1. How many such sequences of positive integers are there?

30. Three mutually tangent spheres each with radius r = 5 sit on a horizontal plane. A triangular pyramid has a base that is an equilateral triangle with side length 6, has three congruent isosceles triangles for vertical faces, and has height h = 12. The base of the pyramid is parallel to the plane, and the vertex of the pyramid is pointing downward so that it is between the base and the plane. Each of the three vertical faces of the pyramid is tangent to one of the spheres at a point on the triangular face along its altitude from the vertex of the pyramid to the side of length 6. The distance that these points of tangency are from the base of the pyramid is m/n, where m and n are relatively prime positive integers. Find m + n.

 

[Deleted member 4993]

26. Let ABCD be a cyclic quadrilateral with AB = 1, BC = 2, CD = 3, and DA = 4. Find the square of the area of the quadrilateral ABCD.

What is the definition of cyclic quadrilateral?

The area of a cyclic quadrilateral follows a similar formula to that of Heron's formula for area of a triangle (Every triangle is a degenerated cyclic quadrilateral with one side length = 0).