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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1962 Swedish Mathematical Competition
1962 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(5)
5
1
Hide problems
max cube inside a regular tetrahedron
Find the largest cube which can be placed inside a regular tetrahedron with side
1
1
1
so that one of its faces lies on the base of the tetrahedron.
4
1
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2 lines intersect in pairs, n=a^4+b^4, a_1 cosx + a_2 cos2x +... +a_n cosnx > 0
Which of the following statements are true? (A)
X
X
X
implies
Y
Y
Y
, or
Y
Y
Y
implies
X
X
X
, where
X
X
X
is the statement, the lines
L
1
,
L
2
,
L
3
L_1, L_2, L_3
L
1
,
L
2
,
L
3
lie in a plane, and
Y
Y
Y
is the statement, each pair of the lines
L
1
,
L
2
,
L
3
L_1, L_2, L_3
L
1
,
L
2
,
L
3
intersect. (B) Every sufficiently large integer
n
n
n
satisfies
n
=
a
4
+
b
4
n = a^4 + b^4
n
=
a
4
+
b
4
for some integers a, b. (C) There are real numbers
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2,... , a_n
a
1
,
a
2
,
...
,
a
n
such that
a
1
cos
x
+
a
2
cos
2
x
+
.
.
.
+
a
n
cos
n
x
>
0
a_1 \cos x + a_2 \cos 2x +... + a_n \cos nx > 0
a
1
cos
x
+
a
2
cos
2
x
+
...
+
a
n
cos
n
x
>
0
for all real
x
x
x
.
3
1
Hide problems
n^2 - 3mn + m - n = 0
Find all pairs
(
m
,
n
)
(m, n)
(
m
,
n
)
of integers such that
n
2
−
3
m
n
+
m
−
n
=
0
n^2 - 3mn + m - n = 0
n
2
−
3
mn
+
m
−
n
=
0
.
2
1
Hide problems
circumradius of PQR, in square ABCD
A
B
C
D
ABCD
A
BC
D
is a square side
1
1
1
.
P
P
P
and
Q
Q
Q
lie on the side
A
B
AB
A
B
and
R
R
R
lies on the side
C
D
CD
C
D
. What are the possible values for the circumradius of
P
Q
R
PQR
PQR
?
1
1
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polynomial f(2x) = f'(x) f''(x)
Find all polynomials
f
(
x
)
f(x)
f
(
x
)
such that
f
(
2
x
)
=
f
′
(
x
)
f
′
′
(
x
)
f(2x) = f'(x) f''(x)
f
(
2
x
)
=
f
′
(
x
)
f
′′
(
x
)
.