Question 1209365: (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|?
Answer by ikleyn(52914)   (Show Source):
You can put this solution on YOUR website! (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|?
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I will solve here part (a), ONLY.
Take a_3 = a_1.
We always can do it.
Then in the equation
|a_1 - a_2| + 2 |a_2 - a_3| + 3 |a_3 - a_1| = 1
the addend 3|a_3 - a_1| will be zero, the addend 2|a_2 - a_3| will become 2|a_2 - a_1|,
and the equation will take the form
3|a_1 - a_2| = 1.
It says that value of |a_1 - a_2| is 1/3 then.
If to think 1 minute, placing mentally a_3 inside the interval [a_1,a_2] of the length 1/3
or outside such an interval, it become clear that such solution gives the optimal choice
and produces the maximum possible value of |a_1 - a_2|.
ANSWER. The largest possible value of |a_1 - a_2| is 1/3.
Solved.
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