(1) Let a_1, a_2, a_3 be real numbers such that
|a_1 - a_2| + 2 |a_2 - a_3| + 3 |a_3 - a_1| = 1.
What is the largest possible value of |a_1 - a_2|?
(2) Let a_1, a_2, a_3, \dots, a_{10} be real numbers such that
|a_1 - a_2| + 2 |a_2 - a_3| + 3 |a_3 - a_4| + 4 |a_4 - a_5 | + \dots + 9 |a_9 - a_{10}| + 10 |a_{10} - a_1| = 1.
What is the largest possible value of |a_1 - a_6|?
For both problems, we can use the triangle inequality to simplify expressions.
(1) Starting with the given equation:
|a_1 - a_2| + 2 |a_2 - a_3| + 3 |a_3 - a_1| = 1.
We have:
|a_1 - a_2| + |a_2 - a_3| + |a_2 - a_3| + |a_3 - a_1| + |a_3 - a_1| + |a_1 - a_2| = 1.
Using the triangle inequality, we know that for any three real numbers, the sum of the magnitudes of any two numbers is greater than or equal to the magnitude of their sum. Applying this to each group of three terms in the above equation, we have:
|a_1 - a_2 + a_2 - a_3| + |a_2 - a_3 + a_3 - a_1| + |a_1 - a_2 + a_3 - a_1| ≥ 1.
Simplifying, we have:
|a_1 - a_3| + |a_2 - a_1| + |a_3 - a_2| ≥ 1.
Since the magnitudes are nonnegative, we can drop the absolute value signs:
(a_1 - a_3) + (a_2 - a_1) + (a_3 - a_2) ≥ 1.
Simplifying further, we have:
a_1 - a_3 + a_2 - a_1 + a_3 - a_2 ≥ 1,
0 ≥ 1.
This is a contradiction, so there are no real solutions for the given equation.
(2) Using a similar approach, we start with the given equation:
|a_1 - a_2| + 2 |a_2 - a_3| + 3 |a_3 - a_4| + 4 |a_4 - a_5 | + \dots + 9 |a_9 - a_{10}| + 10 |a_{10} - a_1| = 1.
We can rewrite this as:
|a_1 - a_2| + |a_2 - a_3| + |a_3 - a_4| + |a_4 - a_5 | + \dots + |a_9 - a_{10}| + |a_{10} - a_1| + |a_2 - a_3| + |a_3 - a_4| + \dots + |a_9 - a_{10}| + |a_{10} - a_1| + \dots + |a_9 - a_{10}| + |a_{10} - a_1| ≥ 1.
Again, using the triangle inequality, we have:
(a_1 - a_2 + a_2 - a_3) + (a_2 - a_3 + a_3 - a_4) + \dots + (a_9 - a_{10} + a_{10} - a_1) + (a_{10} - a_1 + a_1 - a_2) ≥ 1.
This simplifies to:
0 ≥ 1.
Again, this is a contradiction, so there are no real solutions for the given equation.
Therefore, there is no largest possible value of |a_1 - a_2| or |a_1 - a_6| because there are no real solutions for the given equations.
To find the largest possible value of |a_1 - a_2|, we need to analyze the given equation and determine the conditions that maximize this value.
(1) Given equation:
|a_1 - a_2| + 2 |a_2 - a_3| + 3 |a_3 - a_1| = 1
To simplify the analysis, let's consider the absolute values as separate cases based on their signs:
Case 1: a_1 ≤ a_2
In this case, a_1 - a_2 ≤ 0, so we can rewrite the equation as:
-(a_1 - a_2) + 2(a_2 - a_3) + 3(a_3 - a_1) = 1
Simplifying, we get:
2a_2 - a_1 - a_3 = 1
Case 2: a_1 > a_2
In this case, a_1 - a_2 > 0, so we can rewrite the equation as:
(a_1 - a_2) + 2(a_2 - a_3) + 3(a_3 - a_1) = 1
Simplifying, we get:
-2a_1 + 2a_2 + 2a_3 = 1
Now we have two linear equations in terms of a_1, a_2, and a_3. By solving these equations simultaneously, we can find the values of a_1, a_2, and a_3 that satisfy the given equation.
Next, we analyze these equations to find the conditions that maximize |a_1 - a_2|. We want to find the largest possible difference between a_1 and a_2 while still satisfying the given equation. There are two possibilities:
Possibility 1: The largest possible value of a_2 and the smallest possible value of a_1.
This occurs when a_1 - a_2 ≤ 0 (Case 1 above) and a_1 is as small as possible. In this case, the absolute value of a_1 - a_2 is maximized.
Possibility 2: The largest possible value of a_1 and the smallest possible value of a_2.
This occurs when a_1 - a_2 > 0 (Case 2 above) and a_2 is as small as possible. In this case, the absolute value of a_1 - a_2 is also maximized.
To summarize, the largest possible value of |a_1 - a_2| is obtained by considering the two possibilities and finding the maximum value from both cases.
Let's move on to solving the second problem.
(2) Given equation:
|a_1 - a_2| + 2 |a_2 - a_3| + 3 |a_3 - a_4| + 4 |a_4 - a_5 | + \dots + 9 |a_9 - a_{10}| + 10 |a_{10} - a_1| = 1
Similarly, we need to analyze the equation and determine the conditions that maximize |a_1 - a_6|.
To simplify the analysis, let's consider the absolute values as separate cases based on their signs, just like in problem (1).
By following the same steps as in problem (1), we can find the conditions that maximize |a_1 - a_6|. We want to find the largest possible difference between a_1 and a_6 while still satisfying the given equation. By considering the possibilities and finding the maximum value from both cases, we can determine the largest possible value of |a_1 - a_6|.