MathDB
Problems
Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
2010 Finnish National High School Mathematics Competition
2010 Finnish National High School Mathematics Competition
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
5
1
Hide problems
Subset of a plane with a property
Let
S
S
S
be a non-empty subset of a plane. We say that the point
P
P
P
can be seen from
A
A
A
if every point from the line segment
A
P
AP
A
P
belongs to
S
S
S
. Further, the set
S
S
S
can be seen from
A
A
A
if every point of
S
S
S
can be seen from
A
A
A
. Suppose that
S
S
S
can be seen from
A
A
A
,
B
B
B
and
C
C
C
where
A
B
C
ABC
A
BC
is a triangle. Prove that
S
S
S
can also be seen from any other point of the triangle
A
B
C
ABC
A
BC
.
4
1
Hide problems
Football league
In a football season, even number
n
n
n
of teams plays a simple series, i.e. each team plays once against each other team. Show that ona can group the series into
n
−
1
n-1
n
−
1
rounds such that in every round every team plays exactly one match.
3
1
Hide problems
Possible values of P(2005)
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial with integer coefficients and roots
1997
1997
1997
and
2010
2010
2010
. Suppose further that
∣
P
(
2005
)
∣
<
10
|P(2005)|<10
∣
P
(
2005
)
∣
<
10
. Determine what integer values
P
(
2005
)
P(2005)
P
(
2005
)
can get.
2
1
Hide problems
n! has at least 2010 factors
Determine the least
n
∈
N
n\in\mathbb{N}
n
∈
N
such that
n
!
=
1
⋅
2
⋅
3
⋯
(
n
−
1
)
⋅
n
n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n
n
!
=
1
⋅
2
⋅
3
⋯
(
n
−
1
)
⋅
n
has at least
2010
2010
2010
positive factors.
1
1
Hide problems
Medians in right angled triangle
Let
A
B
C
ABC
A
BC
be right angled triangle with sides
s
1
,
s
2
,
s
3
s_1,s_2,s_3
s
1
,
s
2
,
s
3
medians
m
1
,
m
2
,
m
3
m_1,m_2,m_3
m
1
,
m
2
,
m
3
. Prove that
m
1
2
+
m
2
2
+
m
3
2
=
3
4
(
s
1
2
+
s
2
2
+
s
3
2
)
m_1^2+m_2^2+m_3^2=\frac{3}{4}(s_1^2+s_2^2+s_3^2)
m
1
2
+
m
2
2
+
m
3
2
=
4
3
(
s
1
2
+
s
2
2
+
s
3
2
)
.