MathDB
Problems
Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
2014 Finnish National High School Mathematics
2014 Finnish National High School Mathematics
Part of
Finnish National High School Mathematics Competition
Subcontests
(4)
5
1
Hide problems
min pos. integer n = sum of diff. perfect squares, with sum =2014
Determine the smallest number
n
∈
Z
+
n \in Z_+
n
∈
Z
+
, which can be written as
n
=
Σ
a
∈
A
a
2
n = \Sigma_{a\in A}a^2
n
=
Σ
a
∈
A
a
2
, where
A
A
A
is a finite set of positive integers and
Σ
a
∈
A
a
=
2014
\Sigma_{a\in A}a= 2014
Σ
a
∈
A
a
=
2014
. In other words: what is the smallest positive number which can be written as a sum of squares of different positive integers summing to
2014
2014
2014
?
3
1
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broken line of allowed routes between 2 lattice points, not touching y=x line
The points
P
=
(
a
,
b
)
P = (a, b)
P
=
(
a
,
b
)
and
Q
=
(
c
,
d
)
Q = (c, d)
Q
=
(
c
,
d
)
are in the first quadrant of the
x
y
xy
x
y
plane, and
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
are integers satisfying
a
<
b
,
a
<
c
,
b
<
d
a < b, a < c, b < d
a
<
b
,
a
<
c
,
b
<
d
and
c
<
d
c < d
c
<
d
. A route from point
P
P
P
to point
Q
Q
Q
is a broken line consisting of unit steps in the directions of the positive coordinate axes. An allowed route is a route not touching the line
x
=
y
x = y
x
=
y
. Tetermine the number of allowed routes.
2
1
Hide problems
two circumcirlces given, perpendicularity wanted
The center of the circumcircle of the acute triangle
A
B
C
ABC
A
BC
is
M
M
M
, and the circumcircle of
A
B
M
ABM
A
BM
meets
B
C
BC
BC
and
A
C
AC
A
C
at
P
P
P
and
Q
Q
Q
(
P
≠
B
P\ne B
P
=
B
). Show that the extension of the line segment
C
M
CM
CM
is perpendicular to
P
Q
PQ
PQ
.
1
1
Hide problems
x^2+y^2+z^2 if x+y+z=13, xyz= 72, 1/x+1/y+1/z=3/4
Determine the value of the expression
x
2
+
y
2
+
z
2
x^2 + y^2 + z^2
x
2
+
y
2
+
z
2
, if
x
+
y
+
z
=
13
x + y + z = 13
x
+
y
+
z
=
13
,
x
y
z
=
72
xyz= 72
x
yz
=
72
and
1
x
+
1
y
+
1
z
=
3
4
\frac1x + \frac1y + \frac1z = \frac34
x
1
+
y
1
+
z
1
=
4
3
.