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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1980 Swedish Mathematical Competition
1980 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
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min c such that for every 4 points in unit square, exists 2 at distance <c
Find the smallest constant
c
c
c
such that for every
4
4
4
points in a unit square there are two a distance
≤
c
\leq c
≤
c
apart.
5
1
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no of words with 12 letters constructed by 2 symbols
A word is a string of the symbols
a
,
b
a, b
a
,
b
which can be formed by repeated application of the following: (1)
a
b
ab
ab
is a word; (2) if
X
X
X
and
Y
Y
Y
are words, then so is
X
Y
XY
X
Y
; (3) if
X
X
X
is a word, then so is
a
X
b
aXb
a
X
b
. How many words have
12
12
12
letters?
4
1
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int f(x)g(x) dx <= int f(x)g(1-x) dx
The functions
f
f
f
and
g
g
g
are positive and continuous.
f
f
f
is increasing and
g
g
g
is decreasing. Show that
∫
0
1
f
(
x
)
g
(
x
)
d
x
≤
∫
0
1
f
(
x
)
g
(
1
−
x
)
d
x
\int\limits_0^1 f(x)g(x) dx \leq \int\limits_0^1 f(x)g(1-x) dx
0
∫
1
f
(
x
)
g
(
x
)
d
x
≤
0
∫
1
f
(
x
)
g
(
1
−
x
)
d
x
3
1
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T(n+1)-T(n) , if T(n) is the no of dissimilar triangles with integers sides <=n
Let
T
(
n
)
T(n)
T
(
n
)
be the number of dissimilar (non-degenerate) triangles with all side lengths integral and
≤
n
\leq n
≤
n
. Find
T
(
n
+
1
)
−
T
(
n
)
T(n+1)-T(n)
T
(
n
+
1
)
−
T
(
n
)
.
2
1
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|a_i - b_i| not all different, when a_i,b_i permutations of 1-7
a
1
a_1
a
1
,
a
2
a_2
a
2
,
a
3
a_3
a
3
,
a
4
a_4
a
4
,
a
5
a_5
a
5
,
a
6
a_6
a
6
,
a
7
a_7
a
7
and
b
1
b_1
b
1
,
b
2
b_2
b
2
,
b
3
b_3
b
3
,
b
4
b_4
b
4
,
b
5
b_5
b
5
,
b
6
b_6
b
6
,
b
7
b_7
b
7
are two permutations of
1
,
2
,
3
,
4
,
5
,
6
,
7
1, 2, 3, 4, 5, 6, 7
1
,
2
,
3
,
4
,
5
,
6
,
7
. Show that
∣
a
1
−
b
1
∣
|a_1 - b_1|
∣
a
1
−
b
1
∣
,
∣
a
2
−
b
2
∣
|a_2 - b_2|
∣
a
2
−
b
2
∣
,
∣
a
3
−
b
3
∣
|a_3 - b_3|
∣
a
3
−
b
3
∣
,
∣
a
4
−
b
4
∣
|a_4 - b_4|
∣
a
4
−
b
4
∣
,
∣
a
5
−
b
5
∣
|a_5 - b_5|
∣
a
5
−
b
5
∣
,
∣
a
6
−
b
6
∣
|a_6 - b_6|
∣
a
6
−
b
6
∣
,
∣
a
7
−
b
7
∣
|a_7 - b_7|
∣
a
7
−
b
7
∣
are not all different.
1
1
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log_{10} 2 is irrational.
Show that
log
10
2
\log_{10} 2
lo
g
10
2
is irrational.