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Algebra Level 4

Positive integers a , b , c a,b,c are chosen such that a < b < c a<b<c and a b , a c , b c ab,ac,bc forms an increasing arithmetic sequence. What is the minimum value of a b c abc ?


The answer is 36.

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Daniel Wang by Daniel Wang , 22, USA

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2 solutions

Dheeraj Mohan
Jan 5, 2015

ab, ac, and bc are in A.P, This implies that (ab+bc)/2 = ac. b (a+c)/2 = ac Through random guessing of values of a and c, We find that a=2, c=6, b=3 abc=36

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Is there no systematic method of doing it? I was able to see that a a , b b and c c are in harmonic progression. Any way to proceed?

Deeparaj Bhat - 5 years, 3 months ago
Ritabrata Roy
Aug 11, 2018

The expression ,we need from 2ac=ab+bc is 

       2/b=(a+c)/ac

    note a+c=2k(k is natural)

    it implies ac=bk.

Surely k=1 don't access to answer. A little notice about k=2 is also unacceptable. The next k=3 is our guest. We get a=2,b=3,c=6

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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