MathDB
Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1999 Canada National Olympiad
4
4
Part of
1999 Canada National Olympiad
Problems
(1)
17 numbers
Source: Canada 1999
3/4/2006
Suppose
a
1
,
a
2
,
⋯
,
a
8
a_1,a_2,\cdots,a_8
a
1
,
a
2
,
⋯
,
a
8
are eight distinct integers from
{
1
,
2
,
⋯
,
16
,
17
}
\{1,2,\cdots,16,17\}
{
1
,
2
,
⋯
,
16
,
17
}
. Show that there is an integer
k
>
0
k > 0
k
>
0
such that the equation
a
i
−
a
j
=
k
a_i - a_j = k
a
i
−
a
j
=
k
has at least three different solutions. Also, find a specific set of 7 distinct integers from
{
1
,
2
,
…
,
16
,
17
}
\{1,2,\ldots,16,17\}
{
1
,
2
,
…
,
16
,
17
}
such that the equation
a
i
−
a
j
=
k
a_i - a_j = k
a
i
−
a
j
=
k
does not have three distinct solutions for any
k
>
0
k > 0
k
>
0
.
combinatorics