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Problems
Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
2004 Finnish National High School Mathematics Competition
2004 Finnish National High School Mathematics Competition
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
4
1
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Seqeunce with exactly 12 primes
The numbers
2005
!
+
2
,
2005
!
+
3
,
.
.
.
,
2005
!
+
2005
2005! + 2, 2005! + 3, ... , 2005! + 2005
2005
!
+
2
,
2005
!
+
3
,
...
,
2005
!
+
2005
form a sequence of
2004
2004
2004
consequtive integers, none of which is a prime number. Does there exist a sequence of
2004
2004
2004
consequtive integers containing exactly
12
12
12
prime numbers?
3
1
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Again externally tangent circles
Two circles with radii
r
r
r
and
R
R
R
are externally tangent. Determine the length of the segment cut from the common tangent of the circles by the other common tangents.
2
1
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A fraction is an integer
a
,
b
a, b
a
,
b
and
c
c
c
are positive integers and
a
3
+
b
b
3
+
c
\frac{a\sqrt{3} + b}{b\sqrt{3} + c}
b
3
+
c
a
3
+
b
is a rational number. Show that
a
2
+
b
2
+
c
2
a
+
b
+
c
\frac{a^2 + b^2 + c^2}{a + b + c}
a
+
b
+
c
a
2
+
b
2
+
c
2
is an integer.
5
1
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Minimal number of coins
Finland is going to change the monetary system again and replace the Euro by the Finnish Mark. The Mark is divided into
100
100
100
pennies. There shall be coins of three denominations only, and the number of coins a person has to carry in order to be able to pay for any purchase less than one mark should be minimal. Determine the coin denominations.
1
1
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Quadratic equations
The equations
x
2
+
2
a
x
+
b
2
=
0
x^2 +2ax+b^2 = 0
x
2
+
2
a
x
+
b
2
=
0
and
x
2
+
2
b
x
+
c
2
=
0
x^2 +2bx+c^2 = 0
x
2
+
2
b
x
+
c
2
=
0
both have two diff erent real roots. Determine the number of real roots of the equation
x
2
+
2
c
x
+
a
2
=
0.
x^2 + 2cx + a^2 = 0.
x
2
+
2
c
x
+
a
2
=
0.