1990 AHSME Problems/Problem 26
Problem
Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to them in the circle. Then each person computes and announces the average of the numbers of their two neighbors. The figure shows the average announced by each person (not the original number the person picked.) The number picked by the person who announced the average was
Solution 1 (Ten Variables)
For suppose Person picks the number and announces the number We wish to find
Taking the indices modulo we are given that from which
We have ten equations: five with odd-numbered indices and five with even-numbered indices. Note that these two sets of equations are independent. The set that involves is Summing these five equations, we get from which Subtracting from we obtain
~Misof (Solution)
~MRENTHUSIASM (Revision)
Solution 2 (One Variable)
For suppose Person announces the number
Let be the number picked by Person We construct the following table: We have from which
~MRENTHUSIASM
See also
1990 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 25 |
Followed by Problem 27 | |
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