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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1981 Swedish Mathematical Competition
1981 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
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triangles with sides a prime, b power of 2, c square of an odd integer
Show that there are infinitely many triangles with side lengths
a
a
a
,
b
b
b
,
c
c
c
, where
a
a
a
is a prime,
b
b
b
is a power of
2
2
2
and
c
c
c
is the square of an odd integer.
5
1
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area inequality (XYZ) >= (AYZ), (BZX), (CXY)
A
B
C
ABC
A
BC
is a triangle.
X
X
X
,
Y
Y
Y
,
Z
Z
Z
lie on
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
respectively. Show that area
X
Y
Z
XYZ
X
Y
Z
cannot be smaller than each of area
A
Y
Z
AYZ
A
Y
Z
, area
B
Z
X
BZX
BZX
, area
C
X
Y
CXY
CX
Y
.
4
1
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cube of side 5is divided into 125 black and white unit cubes
A cube side
5
5
5
is divided into
125
125
125
unit cubes.
N
N
N
of the small cubes are black and the rest white. Find the smallest
N
N
N
such that there must be a row of
5
5
5
black cubes parallel to one of the edges of the large cube.
3
1
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degree 5 pol, p(x) + 1 is divisible by (x-1)^3, p(x)-1 divisible by (x+1)^3
Find all polynomials
p
(
x
)
p(x)
p
(
x
)
of degree
5
5
5
such that
p
(
x
)
+
1
p(x) + 1
p
(
x
)
+
1
is divisible by
(
x
−
1
)
3
(x-1)^3
(
x
−
1
)
3
and
p
(
x
)
−
1
p(x) - 1
p
(
x
)
−
1
is divisible by
(
x
+
1
)
3
(x+1)^3
(
x
+
1
)
3
.
2
1
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3x3 system, x^y = z , y^z = x, z^x = y
Does
{
x
y
=
z
y
z
=
x
z
x
=
y
\left\{ \begin{array}{l} x^y = z \\ y^z = x \\ z^x = y \\ \end{array} \right.
⎩
⎨
⎧
x
y
=
z
y
z
=
x
z
x
=
y
have any solutions in positive reals apart from
x
=
y
=
z
=
1
x = y = z= 1
x
=
y
=
z
=
1
?
1
1
Hide problems
11... 122 ...25 is a perfect square
Let
N
=
11
⋯
122
⋯
25
N = 11\cdots 122 \cdots 25
N
=
11
⋯
122
⋯
25
, where there are
n
n
n
1
1
1
s and
n
+
1
n+1
n
+
1
2
2
2
s. Show that
N
N
N
is a perfect square.