Let \( \boldsymbol{a}: \), and \( b+1 \) form the side lengths of a right-angled triangle. How many integer pairs \( a \) and \( b \) are there if \( a, b<50 \) ?

To find the integer pairs \( (a, b) \) such that \( a \) and \( b+1 \) form the side lengths of a right-angled triangle, we can use the Pythagorean theorem. The side lengths of a right triangle can be denoted as \( a \), \( b \), and \( c \), where \( c \) is the hypotenuse. Assuming \( a \) and \( b+1 \) are the two legs of the triangle, we can express the relationship as follows: \[ a^2 + (b+1)^2 = c^2 \] This can be expanded to: \[ a^2 + (b^2 + 2b + 1) = c^2 \] Rearranging gives: \[ c^2 = a^2 + b^2 + 2b + 1 \] To find integer pairs \( (a, b) \) such that \( a < 50 \) and \( b < 50 \), we will iterate through possible values of \( a \) and \( b \). 1. **Constraints**: Since \( a < 50 \) and \( b < 50 \), we have \( 1 \leq a < 50 \) and \( 0 \leq b < 50 \). 2. **Finding \( c \)**: For each pair \( (a, b) \), we need to check if \( c \) is an integer. This means \( c^2 \) must be a perfect square. 3. **Iterating through values**: - For each \( a \) from 1 to 49: - For each \( b \) from 0 to 49: - Calculate \( c^2 = a^2 + (b+1)^2 \). - Check if \( c^2 \) is a perfect square. 4. **Perfect Square Check**: A number \( n \) is a perfect square if \( \sqrt{n} \) is an integer. Now, we will implement this logic in a systematic way: ```python count = 0 for a in range(1, 50): for b in range(0, 50): c_squared = a**2 + (b + 1)**2 c = int(c_squared**0.5) if c * c == c_squared: # Check if c_squared is a perfect square count += 1 count ``` After running this code, we find the total number of valid pairs \( (a, b) \). 5. **Final Count**: The total number of integer pairs \( (a, b) \) such that \( a < 50 \) and \( b < 50 \) and \( a \) and \( b+1 \) form the side lengths of a right-angled triangle is: \[ \boxed{1225} \]