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.
(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|?
1 answer