Find Positive Integer Solution for a Given Equation
The problem provides a hidden monotonically increasing function f(x, y)
, and the task is to find all pairs (x, y)
such that f(x, y) == z
.
Here’s a simple approach:
 Initialize the Result: Create an empty result list to store the pairs
(x, y)
that satisfy the condition.  Iterate Through Possible x Values: Since the constraints provide
1 <= x, y <= 1000
, we can iterate through all possible values ofx
from 1 to 1000. a. Find the Corresponding y Value: For eachx
, start withy
as 1 and keep incrementingy
untilf(x, y) <= z
. Iff(x, y) == z
, add the pair(x, y)
to the result. b. Take Advantage of Monotonicity: Sincef(x, y)
is monotonically increasing, once we findf(x, y) > z
, we can break the loop fory
as the further values will only increase.  Return the Result: Return the result list containing the pairs.
Here’s the code:


This code efficiently leverages the monotonically increasing property of the function and the constraints to find the required pairs (x, y)
. By breaking the inner loop when f(x, y) > z
, we avoid unnecessary calculations.