Bunuel wrote:

ABCD is a quadrilateral, as shown in the diagram below. What is the minimum possible sum of areas of triangles AOD and BOC?

(1) Area of triangles AOB and COD are 16 and 36, respectively.

(2) Area of triangle AOD is equal to the area of triangle BOC.

Attachment:

2017-07-24_1034.png

(1) We have \(S_{AOB} = 16\) and \(S_{COD} = 36\)

Also \(\frac{S_{AOB}}{S_{BOC}}=\frac{AO}{OC}\)

\(\frac{S_{AOD}}{S_{DOC}}=\frac{AO}{OC}\)

Hence \(\frac{S_{AOB}}{S_{BOC}} = \frac{S_{AOD}}{S_{DOC}} \\

\implies S_{AOD} * S_{BOC} = S_{AOB} *S_{DOC} = 16 * 36 \)

ALso \(16 * 36 = S_{AOD} * S_{BOC} \leq \frac{1}{4}( S_{AOD} + S_{BOC})^2 \implies S_{AOD} + S_{BOC} \geq 48\)

Sufficient.

(2) We have no more information about the area of AOD and BOC, hence we can't know the minimum possible sum of areas of triangles AOD and BOC. Insufficient.

The answer is A.