MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1975 Swedish Mathematical Competition
1975 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
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|f'(x)| <= C |f(x)|
f
(
x
)
f(x)
f
(
x
)
is defined for
0
≤
x
≤
1
0 \leq x \leq 1
0
≤
x
≤
1
and has a continuous derivative satisfying
∣
f
′
(
x
)
∣
≤
C
∣
f
(
x
)
∣
|f'(x)| \leq C|f(x)|
∣
f
′
(
x
)
∣
≤
C
∣
f
(
x
)
∣
for some positive constant
C
C
C
. Show that if
f
(
0
)
=
0
f(0) = 0
f
(
0
)
=
0
, then
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
for the entire interval.
5
1
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n divides 2^n + 1 for infinitely many positive integers n
Show that
n
n
n
divides
2
n
+
1
2^n + 1
2
n
+
1
for infinitely many positive integers
n
n
n
.
4
1
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concurrency wanted, parallels and equal distances related
P
1
P_1
P
1
,
P
2
P_2
P
2
,
P
3
P_3
P
3
,
Q
1
Q_1
Q
1
,
Q
2
Q_2
Q
2
,
Q
3
Q_3
Q
3
are distinct points in the plane. The distances
P
1
Q
1
P_1Q_1
P
1
Q
1
,
P
2
Q
2
P_2Q_2
P
2
Q
2
,
P
3
Q
3
P_3Q_3
P
3
Q
3
are equal.
P
1
P
2
P_1P_2
P
1
P
2
and
Q
2
Q
1
Q_2Q_1
Q
2
Q
1
are parallel (not antiparallel), similarly
P
1
P
3
P_1P_3
P
1
P
3
and
Q
3
Q
1
Q_3Q_1
Q
3
Q
1
, and
P
2
P
3
P_2P_3
P
2
P
3
and
Q
3
Q
2
Q_3Q_2
Q
3
Q
2
. Show that
P
1
Q
1
P_1Q_1
P
1
Q
1
,
P
2
Q
2
P_2Q_2
P
2
Q
2
and
P
3
Q
3
P_3Q_3
P
3
Q
3
intersect in a point.
3
1
Hide problems
a^n + b^n + c^n >= ab^{n-1} + bc^{n-1} + ca^{n-1}
Show that
a
n
+
b
n
+
c
n
≥
a
b
n
−
1
+
b
c
n
−
1
+
c
a
n
−
1
a^n + b^n + c^n \geq ab^{n-1} + bc^{n-1} + ca^{n-1}
a
n
+
b
n
+
c
n
≥
a
b
n
−
1
+
b
c
n
−
1
+
c
a
n
−
1
for real
a
,
b
,
c
≥
0
a,b,c \geq 0
a
,
b
,
c
≥
0
and
n
n
n
a positive integer.
2
1
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fractional part of = (3+\sqrt{5} )^n >0.99
Is there a positive integer
n
n
n
such that the fractional part of
(
3
+
5
)
n
>
0.99
?
\left(3+\sqrt{5}\right)^n >0.99 ?
(
3
+
5
)
n
>
0.99
?
1
1
Hide problems
AQ_|_QP , A(1,0), L lies on line y = kx
A
A
A
is the point
(
1
,
0
)
(1,0)
(
1
,
0
)
,
L
L
L
is the line
y
=
k
x
y = kx
y
=
k
x
(where
k
>
0
k > 0
k
>
0
). For which points
P
(
t
,
0
)
P(t,0)
P
(
t
,
0
)
can we find a point
Q
Q
Q
on
L
L
L
such that
A
Q
AQ
A
Q
and
Q
P
QP
QP
are perpendicular?