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Contests
National and Regional Contests
Russia Contests
Soros Olympiad in Mathematics
VI Soros Olympiad 1999 - 2000 (Russia)
8.8
8.8
Part of
VI Soros Olympiad 1999 - 2000 (Russia)
Problems
(1)
S_n + 1 is divisible by 2^{n-2} (VI Soros Olympiad 1990-00 R1 8.8)
Source:
5/27/2024
Let
p
1
p_1
p
1
,
p
2
p_2
p
2
,
.
.
.
...
...
,
p
n
p_n
p
n
be different prime numbers (
n
≥
2
n\ge 2
n
≥
2
). All possible products containing an even number of coefficients (all coefficients are different) are composed of these numbers. Let
S
n
S_n
S
n
be the sum of all such products. For example,
S
4
=
p
1
p
2
+
p
1
p
3
+
p
1
p
4
+
p
2
p
3
+
p
2
p
4
+
p
3
p
4
+
p
1
p
2
p
3
p
4
.
S_4 = p_1p_2 + p_1p_3 + p_1p_4 + p_2p_3 + p_2p_4 + p_3p_4+ p_1p_2p_3p_4.
S
4
=
p
1
p
2
+
p
1
p
3
+
p
1
p
4
+
p
2
p
3
+
p
2
p
4
+
p
3
p
4
+
p
1
p
2
p
3
p
4
.
Prove that
S
n
+
1
S_n + 1
S
n
+
1
is divisible by
2
n
−
2
2^{n-2}
2
n
−
2
.
number theory