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Ireland National Math Olympiad
2023 Irish Math Olympiad
P2
P2
Part of
2023 Irish Math Olympiad
Problems
(1)
Special n-triangles
Source: Irish Mathematical Olympiad 2023 Problem 2
5/15/2023
For
n
≥
3
n \geq 3
n
≥
3
, a special n-triangle is a triangle with
n
n
n
distinct numbers on each side such that the sum of the numbers on a side is the same for all sides. For instance, because
41
+
23
+
43
=
43
+
17
+
47
=
47
+
19
+
41
41 + 23 + 43 = 43 + 17 + 47 = 47 + 19 + 41
41
+
23
+
43
=
43
+
17
+
47
=
47
+
19
+
41
, the following is a special
3
3
3
-triangle:
41
41
41
23
19
23\text{ }\text{ }\text{ }\text{ }\text{ }19
23
19
43
17
47
43\text{ }\text{ }\text{ }\text{ }\text{ }17\text{ }\text{ }\text{ }\text{ }\text{ }47
43
17
47
Note that a special
n
n
n
-triangle contains
3
(
n
−
1
)
3(n - 1)
3
(
n
−
1
)
numbers.An infinite set
A
A
A
of positive integers is a special set if, for each
n
≥
3
n \geq 3
n
≥
3
, the smallest
3
(
n
−
1
)
3(n - 1)
3
(
n
−
1
)
numbers of
A
A
A
can be used to form a special
n
n
n
-triangle.Show that the set of positive integers that are not multiples of
2023
2023
2023
is a special set.
combinatorics