Monthly Archives: February 2016

Root-Mean-Square| Primitive Triangle(a,b,c) such that a^2 + c^2 = 2b^2

  Root-Mean-Square http://mathworld.wolfram.com/Root-Mean-Square.html   parametric representation of all primitive solutions     of the Diophantine equation: ,     where     are relatively prime, with     and     an integer     Perimeter where   : OR   … Continue reading

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(x-n)^3 + (x-n+1)^3 + … + x^3 + (x+1)^3 + … + (x+n)^3

  An integer of the form     is called a Triangular number.           In general:                                    

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Heronian triangles with one side Even of length a = 4,6,8,10 –(Part 2)

    Heronian triangles with one side odd of length a = 3,5,7,9,11    (Part 1)   There are infinitely many of primitive Heronian triangles with one side of length   a > 2     If     is … Continue reading

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Heronian triangles with one side odd of length a = 3,5,7,9,11

  There are infinitely many of primitive Heronian triangles with one side of length a > 2   if     is odd : the other sides are:     ,     where     is a solution of the Pelian … Continue reading

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Puzzle | convex quadrilateral

    Find a convex quadrilateral such that the quadrilateral has the smallest area and all four of its sides (a, b, c, d),   two diagonals (e, f) and area K are : (1)   distinct integers (2)   … Continue reading

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Nested trigonometric functions: sine and cosine

      and   Why is     always greater?   And,     Does the ratio     depend on   ?                                 … Continue reading

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sin^2(θ+α) + sin^2(θ+β) – sin^2(α-β) = 2cos(α-β)sin(θ+α)sin(θ+β)

    Prove that                                      

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Median AD in any triangle ABC

    Prove that in any triangle   ,   where     is a median and                                        

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Diophantine equation: (x^8 + y^8 + z^8) = 2(x^16 + y^16 + z^16)

    Find integer solutions to                                    

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A 4×4 matrix containing distinct integers ≤ 27

    Here’s a 4×4 matrix that contains distinct integers ≤ 27         whose row sums are the same : 1 + 2 + 18 + 24 = 45 8 + 15 + 10 + 12 = … Continue reading

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