+2729 (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|?
+410 1)
To simplify this equation, we want to eliminate some of the terms. Set
x=a1−a2
y=a2−a3
Therefore:
a3−a1=−(a1−a3)=−(x+y)=−x−y.
Now our equation is:
|x|+2|y|+3|−x−y|=1.
|x|+2|y|+3|x+y|=1.
One way to approach this is by graphing.
We split this into 3 sections:
One for x≥0,x<0
y≥0,y<0
For |x+y| we have
x>−y,x<−y.
Here is a graph, with x=0,y=0,x=−y labeled. (The green one is |x|+2|y|+3|x+y|=1).
We see the largest possible value of |x| in this graph is 13.
Therefore, the maximum value is |a1−a2|=13.
+410 1)
To simplify this equation, we want to eliminate some of the terms. Set
x=a1−a2
y=a2−a3
Therefore:
a3−a1=−(a1−a3)=−(x+y)=−x−y.
Now our equation is:
|x|+2|y|+3|−x−y|=1.
|x|+2|y|+3|x+y|=1.
One way to approach this is by graphing.
We split this into 3 sections:
One for x≥0,x<0
y≥0,y<0
For |x+y| we have
x>−y,x<−y.
Here is a graph, with x=0,y=0,x=−y labeled. (The green one is |x|+2|y|+3|x+y|=1).
We see the largest possible value of |x| in this graph is 13.
Therefore, the maximum value is |a1−a2|=13.
