Goldman Sachs Interview Question: a, b, c are integers. Such th... | Glassdoor

## Interview Question

Quantitative Analyst Interview London, England (UK)

# a, b, c are integers. Such that a^2 + 2bc = 1; b

^2 + 2ac=2012, find all the possibles values of c^2 + 2ab.

5

2011 is prime

Interview Candidate on Mar 21, 2012
3

More details in addition to the last post.
1) subtracting the 1st equation from the 2nd one gives:
(b-a)*(b+a-2c)＝2011
2) 2011 is prime, so there are only two possibilities:
b-a=1, b+a-2c=2011; or
b-a=2011, b+a-2c=1
3) subtracting the 1st equation from the 3rd equation gives:
(c-a)(c+a-2b)=x-1
4) plugging c-a=0.5*[(b-a)-(b+a-2c)], c+a-2b=(c-a)-2(b-a) into 3) gives you x

Candidate on May 4, 2013
0

In 2) there should be four rather than two possibilities. Both terms could be negative.

Candidate on May 4, 2013
0

2 cand:
heavy one... I have no idea how did you figure out that 2011 is prime! Do you remember the table of prime numbers from 2 to 10000??

I easily found trivial solution b=0 a=+-1, and then got stucked.
however, later I found that this is unique solution, and (3)=c^2= 1006^2.

I've used your approach. You've got a 4 possibilities, +-1 +-2011 and +-2011 +-1, but 2 of them lead to complex roots. You see, sum (1)+(2)+(3) gives you (a+b+c)^2=1+2012+x, thus x&gt;=-2013, so (b-a) may be only +-1, not +-2011.

Diman on Sep 2, 2013
1

In the equation a^2 + 2bc = 1, obviously, a is either 1 or -1 and either b or c is zero. Then if you derive b from the second equation, b = sqrt(2012 - 2ac), c can't be zero, because sqrt(2012) is not an integer. Therefore b = 0. So, in order for b to be 0, 2012 - 2ac should be zero or 2ac = 2012. =&gt; it's either a = -1 and c = -606 or a =1 and c = 606. In any case, because b is zero, 2bc term in the 3rd equation disappears and c^2 gives an answer 367236.

Anonymous on Feb 20, 2017
0

To the comment above, 2ab term in the 3rd equation, of course

Anonymous on Feb 20, 2017
1

simple eq:
(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca.
from given eq, we know:
(a+b+c)^2 = (2012 + 1 + UNKNOWN) = K^2

with K being anything larger than 45.
45*45 = 2025
46*46 = 2116 ..
so you can figure out the missing term by: (K^2 - 2013)

jw on Aug 29, 2017