Problem 11 · 2020 Math Kangaroo
Medium
Number Theory
Logic & Word Problems
sum-constraintcasework
The circles in the figure are to be numbered from 0 to 10, each with a different number. The five sums of the three numbers along each diameter must all be odd. If one of these sums is as small as possible, what is the largest possible value of one of the remaining sums?
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Answer: E — 21
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Hint 1 of 2
The centre circle is shared by all five diameters; for every diameter-sum to be odd, think about the parity the centre forces.
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Hint 2 of 2
Make one sum smallest by surrounding it with tiny numbers, which pushes the leftover large numbers onto another diameter to maximise it.
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Approach: use parity of the shared centre, then push extremes apart
- Numbers 0..10 are placed; each diameter sums two ends plus the shared centre, and all five sums are odd.
- The shared centre fixes a parity pattern for the diameter ends.
- Putting the smallest numbers on one diameter leaves the large numbers for another; maximising that one gives a sum of 21.
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