Let a, b, c, d, and e be positive integers. The sum of the four numbers on each of the five segments connecting "points" of the star is 28. What is the value of the sum a + b + c + d + e?


 Jun 19, 2021

We'll set up an equation by "counting the same thing in two different ways."  Consider the process of tracing out the star by following the five segments in this order a-c, c-e, e-b, b-d, d-a and adding up all the numbers we encountered in the process.


One way of adding the numbers goes like this: Each time we draw a segment we encounter four numbered points.   The picture clearly shows that each numbered point is the intersection of two segments.  So each number is added to the total exactly twice.  So the total sum we get is 2(a+b+c+d+e) + 2(2+3+4+5+6).


Another way of adding the numbers goes like this: We draw the stars by drawing five segments and each segment encountered contains four numbers adding up to 28.   So the total sum we get is 5*28.


The sums we get in two different ways must be the same, so now we can set up an equation

    2*(a+b+c+d+e) + 2*(2+3+4+5+6) = 5*28.

Solving, we get

    a+b+c+d+e = (5*28 - 2*(2+3+4+5+6)) / 2.

 Jun 20, 2021

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