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Tuesday, April 3, 2012

Mathematical Reflections 2012, Issue 1 - Problem J217

Problem:
If a, b, c are integers such that a^2 + 2bc = 1 and b^2 + 2ca = 2012, find all possible values
of c^2 + 2ab.

Proposed by Titu Andreescu.

Solution:
Subtracting the two given equations, we have (b-a)(b+a-2c)=2011, so we get \left\{ \begin{array}{lll} b & = & a \pm 1 \\ 2c & = & b+a \mp 2011 \end{array} \right. \qquad \left\{ \begin{array}{lll} b & = & a \pm 2011 \\ 2c & = & b+a \mp 1. \end{array} \right.
Substituting these values into the equation a^2+2bc=1, we obtain the four equations

(i) 3a^2-2008a-2011=0
(ii) 3a^2+2008a-2011=0
(iii) 3a^2+6032a+2010\cdot2011-1=0
(iv) 3a^2-6032a+2010\cdot2011-1=0.

Equation (i) gives (a+1)(3a-2011)=0, so a=-1, b=0, c=-1006.
Equation (ii) gives (a-1)(3a+2011)=0, so a=1, b=0, c=1006.
Equations (iii) and (iv) have no solution since their discriminant is negative.
In conclusion, c^2+2ab=1006^2=1012036.

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