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Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1974 Dutch Mathematical Olympiad
1974 Dutch Mathematical Olympiad
Part of
Dutch Mathematical Olympiad
Subcontests
(5)
5
1
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n+1 points, n of them collinear and each pair at integer distances
For every
n
∈
N
n \in N
n
∈
N
, is it possible to make a figure consisting of
n
+
1
n+1
n
+
1
points, where
n
n
n
points lie on one line and one point is not on that line, so that each pair of those points is an integer distance from each other?
1
1
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each of 4 quads inside a unit area convex quad has area < 3/8
A convex quadrilateral with area
1
1
1
is divided into four quadrilaterals divided by connecting the midpoints of the opposite sides. Prove that each of those four quadrilaterals has area
<
3
8
< \frac38
<
8
3
.
4
1
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n^4+6n^3+11n^2+3n+31 a perfect square
For which
n
n
n
is
n
4
+
6
n
3
+
11
n
2
+
3
n
+
31
n^4+6n^3+11n^2+3n+31
n
4
+
6
n
3
+
11
n
2
+
3
n
+
31
a perfect square?
2
1
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4 divides x_1x_2+x_3x_3+...+x_{n-1}x_n+x_nx_1 -n
n
>
2
n>2
n
>
2
numbers,
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ..., x_n
x
1
,
x
2
,
...
,
x
n
are odd . Prove that
4
4
4
divides
x
1
x
2
+
x
2
x
3
+
.
.
.
+
x
n
−
1
x
n
+
x
n
x
1
−
n
.
x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 -n.
x
1
x
2
+
x
2
x
3
+
...
+
x
n
−
1
x
n
+
x
n
x
1
−
n
.
3
1
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| 1/(a+25)- 1/(b+25)| < 1/100
Proove that in every five positive numbers there is a pair, say
a
,
b
a,b
a
,
b
, for which
∣
1
a
+
25
−
1
b
+
25
∣
<
1
100
.
\left| \frac{1}{a+25}- \frac{1}{b+25}\right| <\frac{1}{100}.
a
+
25
1
−
b
+
25
1
<
100
1
.