SOLUTION: 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 $

Question 1188700: 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 $a + b + c + d + e$?
https://latex.artofproblemsolving.com/8/f/3/8f3672db8454d57ff7de0ed65b3e243f7f4e6333.png
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
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?
https://latex.artofproblemsolving.com/8/f/3/8f3672db8454d57ff7de0ed65b3e243f7f4e6333.png
~~~~~~~~~~~~~~~~~~~~

The wording formulation of this problem is TERRIBLE.

(1)   From the wording part, it is unclear, how the numbers are related to / (associated with)
      the star's points/verices.

      Also, it is unclear, what the given numbers 1, 2, 3, 4 and 5 do in this problem.

(2)   This sentence "The sum of the four numbers on each of the five segments" MAKES no SENSE, BECAUSE
      there are NO four numbers in each segment.

(3)   The plot is not adequate to the wording part.

When unprofessional/unprepared person tries to create his own problem, it is usually becomes visible/clear
after the first 3 words.

The feeling is the same as it happens when a nail is driven on glass.

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

From the description of the problem and the figure...

a+1+4+c=28; a+c=23 [1]
b+4+2+d=28; b+d=22 [2]
c+2+5+e=28; c+e=21 [3]
d+5+3+a=28; d+a=20 [4]
e+3+1+b=28; e+b=24 [5]

Adding [1] through [5]...

2(a+b+c+d+e)=110
a+b+c+d+e=55 [6]

ANSWER: a+b+c+d+e=55

Note we can determine the values of a, b, c, d, and e individually, even though the problem doesn't ask us to.

Combining [1], [2], and [6] gives us e=10;
Then using the figure gives us b=14, d=8, a=12, c=11.

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