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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2017 Vietnam National Olympiad
2
2
Part of
2017 Vietnam National Olympiad
Problems
(2)
Minimal polynomial
Source: 2017 VMO Problem 2
1/11/2017
Is there an integer coefficients polynomial
P
(
x
)
P(x)
P
(
x
)
satisfying
{
P
(
1
+
2
3
)
=
1
+
2
3
P
(
1
+
5
)
=
2
+
3
5
\begin{cases} P(1+\sqrt[3]{2})=1+\sqrt[3]{2}\\ P(1+\sqrt{5})=2+3\sqrt{5}\end{cases}
{
P
(
1
+
3
2
)
=
1
+
3
2
P
(
1
+
5
)
=
2
+
3
5
algebra
polynomial
Vietnam National Olympiad 2017 Problem 6
Source:
1/6/2017
Prove that a)
∑
k
=
1
1008
k
C
2017
k
≡
0
\sum_{k=1}^{1008}kC_{2017}^{k}\equiv 0
∑
k
=
1
1008
k
C
2017
k
≡
0
(mod
201
7
2
2017^2
201
7
2
)b)
∑
k
=
1
504
(
−
1
)
k
C
2017
k
≡
3
(
2
2016
−
1
)
\sum_{k=1}^{504}\left ( -1 \right )^kC_{2017}^{k}\equiv 3\left ( 2^{2016}-1 \right )
∑
k
=
1
504
(
−
1
)
k
C
2017
k
≡
3
(
2
2016
−
1
)
(mod
201
7
2
2017^2
201
7
2
)
number theory