Fifteen numbers are written on the circle. Each number is equal to the absolute value of the difference of the next two numbers (moving clockwise). Find the greatest of the numbers, if sum of all numbers is 2.
To solve this problem, we need to find a pattern among the numbers written on the circle. Let's start by assigning variables to the 15 numbers.
Let the numbers be a_1, a_2, a_3, ..., a_14, a_15, written clockwise on the circle.
Given that each number is equal to the absolute value of the difference of the next two numbers, we can express this as:
a_1 = |a_2 - a_3|
a_2 = |a_3 - a_4|
a_3 = |a_4 - a_5|
...
a_14 = |a_15 - a_1|
a_15 = |a_1 - a_2|
Since the sum of all the numbers is given to be 2, we can write the equation:
a_1 + a_2 + a_3 + ... + a_14 + a_15 = 2
Now, let's examine the absolute value equation a_i = |a_(i+1) - a_(i+2)| more closely.
Since the absolute value of any number is always non-negative, we have two cases to consider:
1. a_i = a_(i+1) - a_(i+2) if a_(i+1) - a_(i+2) >= 0
2. a_i = -(a_(i+1) - a_(i+2)) if a_(i+1) - a_(i+2) < 0
Based on these two cases, we can rewrite the equation as:
1. a_i = a_(i+1) - a_(i+2) if a_(i+1) >= a_(i+2)
2. a_i = -(a_(i+1) - a_(i+2)) if a_(i+1) < a_(i+2)
To simplify calculations, we can introduce a new variable b_i = a_i - a_(i+1). In terms of this new variable, we have the following:
1. If b_i >= 0, then a_i = b_i + a_(i+1)
2. If b_i < 0, then a_i = -b_i + a_(i+1)
Now, let's substitute these expressions into the equation for the sum:
(b_1 + a_2) + (b_2 + a_3) + (b_3 + a_4) + ... + (b_14 + a_15) + (b_15 + a_1) = 2
Simplifying further, we get:
b_1 + b_2 + b_3 + ... + b_14 + b_15 + (a_1 + a_2 + a_3 + ... + a_15) = 2
Since the sum of all the numbers is 2, and we have the equation:
a_1 + a_2 + a_3 + ... + a_14 + a_15 = 2
We can substitute this into the equation and simplify again:
b_1 + b_2 + b_3 + ... + b_14 + b_15 + 2 = 2
Now, we have:
b_1 + b_2 + b_3 + ... + b_14 + b_15 = 0
Let's analyze this equation. We know that b_i = a_i - a_(i+1). Considering the 15th number, we have:
b_15 = a_15 - a_1
Substituting this into the equation, we get:
b_1 + b_2 + b_3 + ... + b_14 + (a_15 - a_1) = 0
In other words,
(a_15 - a_1) + (a_1 - a_2) + (a_2 - a_3) + ... + (a_14 - a_15) = 0
Notice that in the sum (a_15 - a_1) cancels out with (a_1 - a_2), (a_2 - a_3) cancels out with (a_3 - a_4), and so on, except for the final term (a_14 - a_15).
This simplifies our equation to:
(a_14 - a_15) = 0
Therefore, a_14 = a_15.
Since the sum of all the numbers is 2, we have the equation:
a_1 + a_2 + a_3 + ... + a_14 + a_15 = 2
Substituting a_14 = a_15, we get:
a_1 + a_2 + a_3 + ... + a_14 + a_14 = 2
2a_14 = 2
Dividing both sides by 2, we find:
a_14 = 1
Thus, the greatest of the 15 numbers is 1.