MathDB

6

Part of 1988 AIME Problems

Problems(1)

Arithmetic sequence square

Source:

3/17/2006
It is possible to place positive integers into the vacant twenty-one squares of the 5×55 \times 5 square shown below so that the numbers in each row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk (*).
[asy] int i; for(i=1; i<5; i=i+1) { draw((0,2*i)--(10,2*i)); draw((2*i,0)--(2*i,10)); } string[] no={"0", "74", "103", "*", "186"}; pair[] yes={(1,1), (3,7), (5,3), (7,9), (9,5)}; for(i=0; i<5; i=i+1) { label(no, yes); } draw(origin--(10,0)--(10,10)--(0,10)--cycle, linewidth(2));[/asy]