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Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
2003 Finnish National High School Mathematics Competition
2003 Finnish National High School Mathematics Competition
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
5
1
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Game with arithmetic progression
Players Aino and Eino take turns choosing numbers from the set
{
0
,
.
.
.
,
n
}
\{0,..., n\}
{
0
,
...
,
n
}
with
n
∈
N
n\in \Bbb{N}
n
∈
N
being fixed in advance. The game ends when the numbers picked by one of the players include an arithmetic progression of length
4.
4.
4.
The one who obtains the progression wins. Prove that for some
n
,
n,
n
,
the starter of the game wins. Find the smallest such
n
.
n.
n
.
4
1
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Diophantine equation
Find pairs of positive integers
(
n
,
k
)
(n, k)
(
n
,
k
)
satisfying
(
n
+
1
)
k
−
1
=
n
!
(n + 1)^k - 1 = n!
(
n
+
1
)
k
−
1
=
n
!
3
1
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Purses on a table
There are six empty purses on the table. How many ways are there to put 12 two-euro coins in purses in such a way that at most one purse remains empty?
2
1
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Integer part of a sequence
Find consecutive integers bounding the expression \frac{1}{x_1 + 1}+\frac{1}{x_2 + 1}+\frac{1}{x_3 + 1}+... +\frac{1}{x_{2001} + 1}+\frac{1}{x_{2002} + 1} where
x
1
=
1
/
3
x_1 = 1/3
x
1
=
1/3
and
x
n
+
1
=
x
n
2
+
x
n
.
x_{n+1} = x_n^2 + x_n.
x
n
+
1
=
x
n
2
+
x
n
.
1
1
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Perpendicularity
The incentre of the triangle
A
B
C
ABC
A
BC
is
I
.
I.
I
.
The rays
A
I
,
B
I
AI, BI
A
I
,
B
I
and
C
I
CI
C
I
intersect the circumcircle of the triangle
A
B
C
ABC
A
BC
at the points
D
,
E
D, E
D
,
E
and
F
,
F,
F
,
respectively. Prove that
A
D
AD
A
D
and
E
F
EF
EF
are perpendicular.