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Friday, April 8, 2016

Mathematical Reflections 2016, Issue 1 - Problem J361

Problem:
Solve in positive integers the equation \dfrac{x^2-y}{8x-y^2}=\dfrac{y}{x}.

Proposed by Titu Andreescu, University of Texas at Dallas, USA

Solution:
The given equation can be retwritten as x^3+y^3+27-9xy=27. Using the well-known identity a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca), we obtain (x+y+3)(x^2+y^2+9-xy-3x-3y)=27. Since x,y are positive integers, then x+y+3 \geq 5 and we have two cases.

(i) x+y+3=9 and x^2+y^2+9-xy-3(x+y)=3. From the first equation we have x+y=6, which gives x^2+y^2=36-2xy. Substituting these values into the second equation, we have (36-2xy)+9-xy-18=3, which gives xy=8. So, we obtain the system of equations
\begin{array}{rcl} x+y&=&6 \\ xy&=&8, \end{array} which gives (x,y) \in \{(2,4),(4,2)\}. An easy check shows that the only solution to the given equation is (x,y)=(4,2).

(ii) x+y+3=27 and x^2+y^2+9-xy-3x-3y=1.  From the first equation we have x+y=24, which gives x^2+y^2=576-2xy. Substituting these values into the second equation, we have (576-2xy)+9-xy-72=3, which gives xy=170. So, we obtain the system of equations \begin{array}{rcl} x+y&=&24 \\ xy&=&170, \end{array} which yields no solutions in positive integers.

In conclusion, (x,y)=(4,2). 

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