Question:
If \( m \) and \( p \) are positive integers and \( m^2 + p^2<100 \), what is the greatest possible value of \( mp \)?
If \( m \) and \( p \) are positive integers and \( m^2 + p^2<100 \), what is the greatest possible value of \( mp \)?
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When dealing with inequalities involving squares and products, try using substitution to simplify the problem and maximize the desired value.
Updated On: Oct 3, 2025
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Solution and Explanation
Step 1: Apply the given condition.
We are given that: \[ m^2 + p^2<100 \quad \text{and} \quad (m - p)^2 + 2mp<100 \] This simplifies to: \[ 2mp<100 - (m - p)^2 \quad \text{(Equation 1)} \] Step 2: Find the condition for maximum value of \( mp \).
When \( (m - p)^2 = 0 \), we get: \[ m = p \] Substituting \( m = p \) into Equation (1): \[ 2mp<100 - 0 \quad \Rightarrow \quad mp<50 \] Step 3: Maximize \( mp \).
Now, we know \( mp<50 \). The greatest integer value for \( mp \) is 49, where \( m = p = 7 \).
Step 4: Conclusion.
The greatest possible value of \( mp \) is 49.
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