Question

Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to him in the circle. Then each person computes and announces the average of the numbers of his 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 $6$ was

$1$

$5$

$6$

$10$

not uniquely determined from the given information

Answer: A

Concepts:recurrence linear system cyclic arrangement

Difficulty rating: 2500

Small Hint:

If $x_i$ is a picked number and $a_i$ the displayed average, then $x_{i-1}+x_{i+1}=2a_i$

Big Hint:

Start with two unknown adjacent picked numbers and propagate around the circle

Solution:

Index the displayed averages $a_0,a_1,\ldots,a_9$ clockwise as $1,2,\ldots,10,$ and let $x_i$ be the corresponding picked numbers. Write $x_0=t,\ x_1=u.$ From $x_{i-1}+x_{i+1}=2a_i,$ successive values are $\begin{aligned} x_2&=4-t, &x_3&=6-u,\\ x_4&=4+t, &x_5&=4+u,\\ x_6&=8-t, &x_7&=10-u,\\ x_8&=8+t, &x_9&=8+u. \end{aligned}$ The two closing equations give $t=6$ and $u=-3.$ Hence $x_5=4+u=1.$

Thus the correct answer is A.

1990 AMC 12 Problem 26

Answer: A

Difficulty rating: 2500

Small Hint:

Big Hint:

Solution:

Problem 26 in Other Years