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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1999 Swedish Mathematical Competition
1999 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
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moves i sequences, replace a,b by their gcd and lcm
S
S
S
is any sequence of at least
3
3
3
positive integers. A move is to take any
a
,
b
a, b
a
,
b
in the sequence such that neither divides the other and replace them by gcd
(
a
,
b
)
(a,b)
(
a
,
b
)
and lcm
(
a
,
b
)
(a,b)
(
a
,
b
)
. Show that only finitely many moves are possible and that the final result is independent of the moves made, except possibly for order.
5
1
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x_1x_2 + x_2x_3 + ... + x_{n-1}x_n <= s^2/4
x
i
x_i
x
i
are non-negative reals.
x
1
+
x
2
+
.
.
.
+
x
n
=
s
x_1 + x_2 + ...+ x_n = s
x
1
+
x
2
+
...
+
x
n
=
s
. Show that
x
1
x
2
+
x
2
x
3
+
.
.
.
+
x
n
−
1
x
n
≤
s
2
4
x_1x_2 + x_2x_3 + ... + x_{n-1}x_n \le \frac{s^2}{4}
x
1
x
2
+
x
2
x
3
+
...
+
x
n
−
1
x
n
≤
4
s
2
.
4
1
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equilateral inscribed in square
An equilateral triangle of side
x
x
x
has its vertices on the sides of a square side
1
1
1
. What are the possible values of
x
x
x
?
1
1
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|||||x^2-x-1| - 2| - 3| - 4| - 5| = x^2 + x - 30
Solve
∣
∣
∣
∣
∣
x
2
−
x
−
1
∣
−
2
∣
−
3
∣
−
4
∣
−
5
∣
=
x
2
+
x
−
30
|||||x^2-x-1| - 2| - 3| - 4| - 5| = x^2 + x - 30
∣∣∣∣∣
x
2
−
x
−
1∣
−
2∣
−
3∣
−
4∣
−
5∣
=
x
2
+
x
−
30
.
2
1
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<OPO' = 90^o wanted, externally tangent circles
Circle
C
C
C
center
O
O
O
touches externally circle
C
′
C'
C
′
center
O
′
O'
O
′
. A line touches
C
C
C
at
A
A
A
and
C
′
C'
C
′
at
B
B
B
.
P
P
P
is the midpoint of
A
B
AB
A
B
. Show that
∠
O
P
O
′
=
9
0
o
\angle OPO' = 90^o
∠
OP
O
′
=
9
0
o
.
3
1
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5^a + 6^b + 7^c + 11^d = 1999
Find non-negative integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
such that
5
a
+
6
b
+
7
c
+
1
1
d
=
1999
5^a + 6^b + 7^c + 11^d = 1999
5
a
+
6
b
+
7
c
+
1
1
d
=
1999
.