Whose rating is higher?

First,Jim solved a problem with rating $1700$ when his current rating is $1600$ ,so his rating increases by

$(\frac{1700-1600}{25})^{2} = (\frac{100}{25})^{2} = 4^{2} = 16$

Jim's rating is now $1616$ .Then, his rating drops by

$(\frac{1591-1616}{25})^{2} = (\frac{-25}{25})^{2} = 1$

So,Jim's rating is now $1615$ .When Bob got his first problem wrong,his rating drops by

$(\frac{1500-1600}{25})^{2} = (\frac{-100}{25})^{2} = 4^{2} = 16$

Bob's rating is $1600 - 16 = 1584$ .To beat Jim,Bob needs to increase his rating by $1615 - 1584 = 31$ .Then,we can construct an inequality

$(\frac{P-1584}{25})^{2} \geq 31$

Taking the square root of both sides, we get

$\frac{P-1584}{25} \geq 5.5677643628......$

Then,we multiply both sides by 25

$P-1584 \geq 139.19410907......$

Adding 1584 to both sides,we get

$P \geq 1723.19410907......$

So our answer is $\boxed{1723}$ .