Problem 21 · 2015 Math Kangaroo
Stretch
Algebra & Patterns
substitutionsum-constraint
In the diagram we can see seven sections which are bordered by three circles. One number is written into each section. It is known that each number is equal to the sum of all the numbers in the adjacent zones. (Two zones are called adjacent if they have more than one corner point in common.) Which number is written into the inner area?
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Answer: A — 0
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Hint 1 of 2
Give each of the seven regions a letter and write 'region = sum of its adjacent regions'.
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Hint 2 of 2
With two outer regions fixed at 1 and 2, solving the linear system pins the centre to 0.
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Approach: solve the linear system of region sums
- Label the seven regions; each equals the sum of the regions sharing an arc with it.
- Two outer regions are 1 and 2; this fixes all the others by the equations.
- Solving, the inner (all-three) region comes out to 0.
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