Problem #7

Algebra Level 1

Find x, given that:

∣ x − 2013 ∣ + ∣ x − 2014 ∣ + ∣ x − 2015 ∣ = 2 \left| x-2013 \right| +\left| x-2014 \right| +\left| x-2015 \right| =2

Authentic test problem - 1st term test - 2014-15 (Hanoi-Amsterdam high school)


The answer is 2014.

Đức Dũng Phạm by Đức Dũng Phạm , 26, Vietnam

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1 solution

There is a mathematical property: ∣ a ∣ + ∣ b ∣ ≥ ∣ a + b ∣ \left| a \right| +\left| b \right| \ge \left| a+b \right| So 2 = ∣ x − 2013 ∣ + ∣ x − 2014 ∣ + ∣ x − 2015 ∣ = ∣ x − 2013 ∣ + ∣ x − 2014 ∣ + ∣ 2015 − x ∣ ≥ ∣ x − 2013 + 2015 − x ∣ + ∣ x − 2014 ∣ ≥ ∣ 2 ∣ + ∣ x − 2014 ∣ ⇒ 0 ≥ ∣ x − 2014 ∣ ⇒ x − 2014 = 0 ⇒ x = 2014 2=\left| x-2013 \right| +\left| x-2014 \right| +\left| x-2015 \right| =\left| x-2013 \right| +\left| x-2014 \right| +\left| 2015-x \right| \\ \ge \left| x-2013+2015-x \right| +\left| x-2014 \right| \\ \ge \left| 2 \right| +\left| x-2014 \right| \\ \Rightarrow 0\ge \left| x-2014 \right| \\ \Rightarrow x-2014=0\\ \Rightarrow x=2014 Note that since ∣ x − 2014 ∣ ≥ 0 \left| x-2014 \right| \ge 0 so we can immediately say x − 2014 = 0 x-2014=0 without proving anything.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

it is easy to guess 2014..

Vighnesh Raut - 6 years, 5 months ago

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Yes. This task is worth 0.5pts/10pts, which is only 5% of the score. And by guessing 2014, you'll get 0.25pts. But what I want to do here is for everyone to think logically about the answer, not to guess. If we do that at class, the teacher will not accept that answer since you haven't proved anything that 2014 is the answer. Anyway, it's easy to guess =))

Đức Dũng Phạm - 6 years, 4 months ago

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