Excellent solutions were sent in by Michael from Wilmslow High School, Cheshire, UK and Andrei from Tudor Vianu National College, Bucharest, Romania.



ratio1 (1) As triangles AOD and AEC are similar
OA
AC
 = OD
EC
 = 2
3
 so OA
OC
 = 2
5
 .
As triangles BOF and BCG are similar
OB
BC
 = OF
GC
 = 3
1
 so OB
OC
 = 3
4
 .
Hence:
AB
OC
 = OB - OA
OC
 = 3
4
 - 2
5
 = 7
20
 .
ratio2 2) From triangles OAD and AXE
OA
AX
 = OD
EX
 = 2
1.5
 = 4
3
and hence
OA
OX
 = 4
7
     and     OA
OC
 = 2
7
 .
Again
OB
OC
 = 3
4
 .
Hence:
AB
OC
 = OB - OA
OC
 = 3
4
 - 2
7
 = 13
28
 .
ratio3 (3) The reasoning in this part is identical to part (1).