How to solve this nice math Exponential algebra problem | Olympiad Question | x=?,y=?

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[Celtigar]

The equation is quadratic in x with discriminant 4y^2+4y-4*144. To admit a real solution, the discriminant must be non-negative, implying that y>=7. When y=7, the quadratic has x=2.2736 as its only positive root, implying that x=1 or x=2. Substituting, we find that only x=2 admits an integer solution x=2, y=8.

[RealQinnMalloryu]

(20)+2xy+(22)=44 (10^10)+2xy+(11^11) (5^5^5^5)+2xy+(5^6^5^6) (3^1^2^1) (32) (xy ➖ 3xy+2).

[ArwindSah]

Easy way to solve this equation
X² + 25XY +Y = 44
X² + 2XY +Y² -Y² + y =44
(X+Y)² - (Y² -Y) =44.

(X+Y)² - (Y² - 2.Y.(1/2)
+(1/2)² +(1/2)² = 44

(X+Y)² -(Y-1/2)²=44 -1/4
(X+Y+Y -1/2)(X+Y -Y+1/2) = 175/4

(X +2Y -1/2)(X +1/2)=175)4
4(X+2Y -1)(X+1/2) =175
(2X +4Y -1)(2X+1) =175
From here you can proceed as you have done.
Thanks