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Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
2001 Finnish National High School Mathematics Competition
2001 Finnish National High School Mathematics Competition
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
5
1
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Divisibility!
Determine
n
∈
N
n \in \Bbb{N}
n
∈
N
such that
n
2
+
2
n^2 + 2
n
2
+
2
divides
2
+
2001
n
.
2 + 2001n.
2
+
2001
n
.
4
1
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Probability in lottery
A sequence of seven digits is randomly chosen in a weekly lottery. Every digit can be any of the digits
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9.
What is the probability of having at most fi ve diff erent digits in the sequence?
3
1
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Inequality with condition an inequality
Numbers
a
,
b
a, b
a
,
b
and
c
c
c
are positive integers and
1
a
+
1
b
+
1
c
<
1.
\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}< 1.
a
1
+
b
1
+
c
1
<
1.
Show that
1
a
+
1
b
+
1
c
≤
41
42
.
\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}\leq \frac{41}{42}.
a
1
+
b
1
+
c
1
≤
42
41
.
2
1
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Parabolas and a line
Equations of non-intersecting curves are
y
=
a
x
2
+
b
x
+
c
y = ax^2 + bx + c
y
=
a
x
2
+
b
x
+
c
and
y
=
d
x
2
+
e
x
+
f
y = dx^2 + ex + f
y
=
d
x
2
+
e
x
+
f
where
a
d
<
0.
ad < 0.
a
d
<
0.
Prove that there is a line of the plane which does not meet either of the curves.
1
1
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Ratio in a right triangle
In the right triangle
A
B
C
,
ABC,
A
BC
,
C
F
CF
CF
is the altitude based on the hypotenuse
A
B
.
AB.
A
B
.
The circle centered at
B
B
B
and passing through
F
F
F
and the circle with centre
A
A
A
and the same radius intersect at a point of
C
B
.
CB.
CB
.
Determine the ratio
F
B
:
B
C
.
FB : BC.
FB
:
BC
.