IMO 1977 LL SWE40

Origin: SWE

Problem

The numbers 1, 2, 3, . . ., 64 are placed on a chessboard, one number in each square. Consider all squares on the chessboard of size 2 \times 2. Prove that there are at least three such squares for which the sum of the 4 numbers contained exceeds 100.

Solution

Let us divide the chessboard into 16 squares Q1, Q2, . . . , Q16 of size 2 \times 2. Let sk be the sum of numbers in Qk, and let us assume that s1 \geqs2 \geq \cdot \cdot \cdot \geqs16. Since s4 + s5 + \cdot \cdot \cdot + s16 \geq1 + 2 + \cdot \cdot \cdot + 52 = 1378, we must have s4 \geq100 and hence s1, s2, s3 \geq100 as well.