Mathematical Olympiad in China
206
elements in S are forbidden. Hence we can select one more element. Remark The size I S I = lo6 is unnecessarily large. The following statement is true: If A is a K -element subset of S = { 1, 2 , , n } and m is a positive integer such that n > (m- 1) ( ( t ) + l ) ,thenthereexisttl, SsuchthatthesetsAj= { x + t j I x E A ) f o r j = l , disjoint.
..a,
..a,
t,E
marepairwise
@ BDetermine all pairs of positive integers (a, b) such that U2
2ab2 - b3
+1
is a positive integer. Solution I Let (a, b) be a pair of positive integers satisfying the
+
> 0, we have 2ab2 -b3 1 > 0 or 2ab2 - b3 1 b 1 a >-, and hence a -.2b Using this, we infer from K 1, or 2 2b2 condition. Since K
U2
=
>
~
u2 >b2
+
(2a - b)
>
+ 1, that u2 > @ (2a
-
a>bor2a=
>0. Hence
b)
0
b.
Now consider the two solutions al , a2 of the equation a2 - 2Kb2a
+ K(b3
-
0
1) = 0
for any fixed positive integers K and b , and assume that one of them is an integer. Then the other is also an integer because a1 +a2 = 2Kb2. We may assume that al a 2 , and we have al Kb2 > 0.
>
Furthermore, since ala2
Together with
= K(b3 - 1)
>
, we get
0, we conclude that a2
=
0 or a2
=
b ( in the latter 2
-