IMO 1992 SL 10

Origin: ITA

Problem

Let V be a finite subset of Euclidean space consisting of points (x, y, z) with integer coordinates. Let S1, S2, S3 be the projections of V onto the yz, xz, xy planes, respectively. Prove that |V |2 \leq|S1||S2||S3| (|X| denotes the number of elements of X).

Solution

Let us set S(x) = {(y, z) | (x, y, z) \inV }, Sy(x) = {z | (x, z) \inSy} and Sz(x) = {y | (x, y) \inSz}. Clearly S(x) \subsetSx and S(x) \subsetSy(x) \times Sz(x). It follows that |V | =  x |S(x)| \leq  x

|Sx||Sy(x)||Sz(x)|

|Sx|  x

|Sy(x)||Sz(x)|. (1) Using the Cauchy–Schwarz inequality we also get  x

|Sy(x)||Sz(x)| \leq ! x |Sy(x)| ! x |Sz(x)| =

|Sy||Sz|. (2) Now (1) and (2) together yield |V | \leq

|Sx||Sy||Sz|.