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Find all positive integer solutions (x, y) of the equation arctan(x) + arccos(y/sqrt(1+y²)) = arcsin(3/sqrt(10)).

1. x = 1, y = 2 only
2. x = 2, y = 7 only
3. x = 1, y = 2 and x = 2, y = 7
4. No positive integer solutions

Correct answer: x = 1, y = 2 and x = 2, y = 7

Solution

Note arccos(y/sqrt(1+y²)) = arctan(1/y) for y > 0, and arcsin(3/sqrt(10)) = arctan(3). So arctan(x) + arctan(1/y) = arctan(3). Taking tangent: (x + 1/y)/(1 - x/y) = 3, i.e. (xy + 1)/(y - x) = 3, so xy + 1 = 3y - 3x, giving xy - 3y + 3x + 1 = 0. Solve: y(x-3) = -3x-1, y = (3x+1)/(3-x). For x=1: y = 4/2 = 2. For x=2: y = 7/1 = 7. x=3 undefined; x>3 gives negative y. So solutions are (1,2) and (2,7).

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