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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1979 Dutch Mathematical Olympiad
1979 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(4)
1
1
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distributions of 6 coins in 4 children
A cent, a stuiver (
5
5
5
cent coin), a dubbeltje (
10
10
10
cent coin), a kwartje (
25
25
25
cent coin), a gulden (
100
100
100
cent coin) and a rijksdaalder (
250
250
250
cent coin) are divided among four children in such a way that each of them receives at least one of the six coins. How many such distributions are there?
4
1
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3 parallel lines, symmetric points wrt angle bisectors
Given is the non-equilateral triangle
A
1
A
2
A
3
A_1A_2A_3
A
1
A
2
A
3
.
B
i
j
B_{ij}
B
ij
is the symmetric of
A
i
A_i
A
i
wrt the inner bisector of
∠
A
j
\angle A_j
∠
A
j
. Prove that lines
B
12
B
21
B_{12}B_{21}
B
12
B
21
,
B
13
B
31
B_{13}B_{31}
B
13
B
31
and
B
23
B
32
B_{23}B_{32}
B
23
B
32
are parallel.
3
1
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last two digits of a_{1979} if a_{n+1} = 9^{a_n}
Define
a
1
=
1979
a_1 = 1979
a
1
=
1979
and
a
n
+
1
=
9
a
n
a_{n+1} = 9^{a_n}
a
n
+
1
=
9
a
n
for
n
=
1
,
2
,
3
,
.
.
.
n = 1,2,3,...
n
=
1
,
2
,
3
,
...
. Determine the last two digits of
a
1979
a_{1979}
a
1979
.
2
1
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a^3=b^3+c^3+12a , a^2=5(b+c), diophantine 2x3 system
Solve in
N
N
N
:
{
a
3
=
b
3
+
c
3
+
12
a
a
2
=
5
(
b
+
c
)
\begin{cases} a^3=b^3+c^3+12a \\ a^2=5(b+c) \end{cases}
{
a
3
=
b
3
+
c
3
+
12
a
a
2
=
5
(
b
+
c
)