very easy algebra

Algebra Level 2

find the arithmetic mean of 9,99,999,9999,99999,999999,9999999,99999999 and 999999999.(there are 9 numbers and they are distinct) Which NUMBER is not appeared in their 9-digit mean?

This problem is from AMC.

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Margaret Zheng by Margaret Zheng , 20, USA

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2 solutions

Chew-Seong Cheong
Feb 26, 2015

The arithmetic mean

μ = 9 + 99 + 999 + 9999 + 99999 + 999999 + 9999999 + 99999999 + 999999999 9 \mu = \frac {9+99+999+9999+99999+999999+9999999+99999999+999999999}{9}

⇒ μ = + 1 + 11 + 111 + 1111 + 11111 + 111111 + 1111111 + 11111111 + 111111111 = 123456789 \Rightarrow \mu = \begin{array} {r} +1 \\ +11 \\ +111 \\ +1111 \\ +11111 \\ +111111 \\ +1111111 \\ +11111111 \\ +111111111 \\ = 123456789 \end{array}

Therefore, number missing is 0 \boxed{0} .

7 Helpful 0 Interesting 0 Brilliant 0 Confused

exactly. :D

Margaret Zheng - 6 years, 3 months ago
Brock Brown
Feb 27, 2015

Python: 

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numbers = []
current = '9'
while len(current) < 10:
    numbers.append(int(current))
    current = current + '9'
mean = sum(numbers)/len(numbers)
for test in xrange(10):
    if str(test) not in str(mean):
        print "Answer:", test

3 Helpful 0 Interesting 0 Brilliant 0 Confused

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