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Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
2013 Finnish National High School Mathematics Competition
2013 Finnish National High School Mathematics Competition
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
5
1
Hide problems
Solve $2^m p^2 + 1 = q^5$
Find all integer triples
(
m
,
p
,
q
)
(m,p,q)
(
m
,
p
,
q
)
satisfying
2
m
p
2
+
1
=
q
5
2^mp^2+1=q^5
2
m
p
2
+
1
=
q
5
where
m
>
0
m>0
m
>
0
and both
p
p
p
and
q
q
q
are prime numbers.
4
1
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Special and superspecial sets
A subset
E
E
E
of the set
{
1
,
2
,
3
,
…
,
50
}
\{1,2,3,\ldots,50\}
{
1
,
2
,
3
,
…
,
50
}
is said to be special if it does not contain any pair of the form
{
x
,
3
x
}
.
\{x,3x\}.
{
x
,
3
x
}
.
A special set
E
E
E
is superspecial if it contains as many elements as possible. How many element there are in a superspecial set and how many superspecial sets there are?
3
1
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Compute AD, where AB is a diameter of (ABC)
The points
A
,
B
,
A,B,
A
,
B
,
and
C
C
C
lies on the circumference of the unit circle. Furthermore, it is known that
A
B
AB
A
B
is a diameter of the circle and
∣
A
C
∣
∣
C
B
∣
=
3
4
.
\frac{|AC|}{|CB|}=\frac{3}{4}.
∣
CB
∣
∣
A
C
∣
=
4
3
.
The bisector of
A
B
C
ABC
A
BC
intersects the circumference at the point
D
D
D
. Determine the length of the
A
D
AD
A
D
.
2
1
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Aina the Algebraist buying tickets
In a particular European city, there are only
7
7
7
day tickets and
30
30
30
day tickets to the public transport. The former costs
7.03
7.03
7.03
euro and the latter costs
30
30
30
euro. Aina the Algebraist decides to buy at once those tickets that she can travel by the public transport the whole three year (2014-2016, 1096 days) visiting in the city. What is the cheapest solution?
1
1
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Determine a monic cubic
The coefficients
a
,
b
,
c
a,b,c
a
,
b
,
c
of a polynomial
f
:
R
→
R
,
f
(
x
)
=
x
3
+
a
x
2
+
b
x
+
c
f:\mathbb{R}\to\mathbb{R}, f(x)=x^3+ax^2+bx+c
f
:
R
→
R
,
f
(
x
)
=
x
3
+
a
x
2
+
b
x
+
c
are mutually distinct integers and different from zero. Furthermore,
f
(
a
)
=
a
3
f(a)=a^3
f
(
a
)
=
a
3
and
f
(
b
)
=
b
3
.
f(b)=b^3.
f
(
b
)
=
b
3
.
Determine
a
,
b
a,b
a
,
b
and
c
c
c
.