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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1995 Swedish Mathematical Competition
1995 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
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signals used for communication are binary sequences of length 10
Signals used for communication are binary sequences of length
10
10
10
. Unfortunately, the receiving device got broken so that it cannot distinguish between two signals unless those differ in more than five places. What is the largest possible number of signals that can still be used to prevent ambiguities?
5
1
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angles of cyclic ABCD wanted
On a circle with center
O
O
O
and radius
r
r
r
are given points
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
in this order such that
A
B
,
B
C
AB, BC
A
B
,
BC
and
C
D
CD
C
D
have the same length
s
s
s
and the length of
A
D
AD
A
D
is
s
+
r
s+ r
s
+
r
.Assume that
s
<
r
s < r
s
<
r
. Determine the angles of quadrilateral
A
B
C
D
ABCD
A
BC
D
.
4
1
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0<a,b<1<c if a,b,c>0 with abc=1, a+b+c>1/a+1/b+1/c
The product of three positive numbers is
1
1
1
and their sum is greater than the sum of their inverses. Prove that one of these numbers is greater than
1
1
1
, while the other two are smaller than
1
1
1
.
3
1
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a+b^2 = x+y^2 and a^2 +b = x^2 +y, if a+b+x+y < 2
Let
a
,
b
,
x
,
y
a,b,x,y
a
,
b
,
x
,
y
be positive numbers with
a
+
b
+
x
+
y
<
2
a+b+x+y < 2
a
+
b
+
x
+
y
<
2
. Given that
{
a
+
b
2
=
x
+
y
2
a
2
+
b
=
x
2
+
y
\begin{cases} a+b^2 = x+y^2 \\ a^2 +b = x^2 +y\end {cases}
{
a
+
b
2
=
x
+
y
2
a
2
+
b
=
x
2
+
y
show that
a
=
x
a = x
a
=
x
and
b
=
y
b = y
b
=
y
2
1
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Botvid left home between 4 and 5 for a short visit to Amanda
Botvid left home between
4
4
4
and
5
5
5
for a short visit to Amanda. When he came back between
5
5
5
and
6
6
6
, he found that the hands of the clock had changed places. What time was it?
1
1
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sum of pages in a mag
All pages of a magazine are numbered and printed on both sides. One sheet with two sides is missing. The numbers of the remaining pages sum to
963
963
963
. How many pages did the magazine have originally and which pages are missing?