Data Sufficiency

IMO D

Statement 1: The coordinates of point R are (0,0) and point S are (5/2,5/2).

Don't fall for the fact that in the diagram RP and RQ seem to be at right angles to each other. Angle RPQ could have a variety of measures. So P and Q can lie at various points along the circle.

As P and Q become closer together, the distance between the origin and S becomes closer to the radius of the circle.

As P and Q become further apart, the difference between the radius of the circle and the distance between S and the origin increases.

So the coordinates of R and S are insufficient for determining the size, or the area, of the circle.

Insufficient.

Statement 2: The coordinates of P and Q are (0,5) and (5,0) respectively.

Once again from the diagram one can get the impression that angle PRQ is a right angle, and so RP and RQ are radii of the circle and are of length 5.

Also, one might be tempted to incorrectly consider information from Statement 1 when evaluating Statement 2, and to thus think of R as on the origin.

However, the truth is that we don't have information indicating that PRQ is a right angle, so the circle could lie in a way such that R is on the origin and RP and RQ are radii of length 5, in a way such that R lies in the second, third or fourth quadrant, in which case the radius would be greater than 5, or in a way such that R lies in the first quadrant, in which case the radius would be less than 5.

Insufficient.

Statements Combined:

If the coordinates of R are (0,0) and the coordinates of P and Q are (0,5) and (5,0) respectively, then we can determine that the lengths of RP and RQ are both 5, giving us the radius of the circle, using which we can calculate the area.

Sufficient.

The correct answer is C.