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National and Regional Contests
Chile Contests
Chile National Olympiad
1998 Chile National Olympiad
7
7
Part of
1998 Chile National Olympiad
Problems
(1)
2 dice with same sum as 2 ordinary ones (Chile NMO 1998 P7)
Source:
11/24/2021
When rolling two normal dice, the set of possible outcomes of the sum of the points is
2
,
3
,
3
,
4
,
4
,
4
,
.
.
.
,
11
,
11
,
12
2, 3, 3, 4,4, 4,..., 11, 11,12
2
,
3
,
3
,
4
,
4
,
4
,
...
,
11
,
11
,
12
. Notice that this sequence can be obtained from the identity
(
x
+
x
2
+
x
3
+
x
4
+
x
5
+
x
6
)
(
x
+
x
2
+
x
3
+
x
4
+
x
5
+
x
6
)
=
x
2
+
2
x
3
+
3
x
4
+
.
.
.
+
2
x
11
+
x
12
.
(x + x^2 + x^3 + x^4 + x^5 + x^6) (x + x^2 + x^3 + x^4 + x^5 + x^6) = x^2 + 2x^3 + 3x^4 +... + 2x^{11} + x^{12}.
(
x
+
x
2
+
x
3
+
x
4
+
x
5
+
x
6
)
(
x
+
x
2
+
x
3
+
x
4
+
x
5
+
x
6
)
=
x
2
+
2
x
3
+
3
x
4
+
...
+
2
x
11
+
x
12
.
Design a crazy pair of dice, that is, two other cubes, not necessarily the same, with a natural number indicated on each face, such that the set of possible results of the sum of its points is equal to of two normal dice.
algebra
combinatorics