Question

If \( m \) and \( p \) are positive integers and \( m^2 + p^2<100 \), what is the greatest possible value of \( mp \)?

Hint:

When dealing with inequalities involving squares and products, try using substitution to simplify the problem and maximize the desired value.

Answer 1

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.