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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1969 Swedish Mathematical Competition
1969 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
3
1
Hide problems
sum a_ib_i <= sum a_i B_i
a
1
≤
a
2
≤
.
.
.
≤
a
n
a_1 \le a_2 \le ... \le a_n
a
1
≤
a
2
≤
...
≤
a
n
is a sequence of reals
b
1
,
b
2
,
b
3
,
.
.
.
,
b
n
b_1, _b2, b_3,..., b_n
b
1
,
b
2
,
b
3
,
...
,
b
n
is any rearrangement of the sequence
B
1
≤
B
2
≤
.
.
.
≤
B
n
B_1 \le B_2 \le ...\le B_n
B
1
≤
B
2
≤
...
≤
B
n
. Show that
∑
a
i
b
i
≤
∑
a
i
B
i
\sum a_ib_i \le \sum a_i B_i
∑
a
i
b
i
≤
∑
a
i
B
i
.
6
1
Hide problems
n triangles from 3n points in the plane
Given
3
n
3n
3
n
points in the plane, no three collinear, is it always possible to form
n
n
n
triangles (with vertices at the points), so that no point in the plane lies in more than one triangle?
5
1
Hide problems
N =a_1a_2...a_n= 0 mod 3 if a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0 mod 3
Let
N
=
a
1
a
2
.
.
.
a
n
N = a_1a_2...a_n
N
=
a
1
a
2
...
a
n
in binary. Show that if
a
1
−
a
2
+
a
3
−
.
.
.
+
(
−
1
)
n
−
1
a
n
=
0
a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0
a
1
−
a
2
+
a
3
−
...
+
(
−
1
)
n
−
1
a
n
=
0
mod
3
3
3
, then
N
=
0
N = 0
N
=
0
mod
3
3
3
.
4
1
Hide problems
min largest value of |y^2 - xy| for y in [0, 1]
Define
g
(
x
)
g(x)
g
(
x
)
as the largest value of
∣
y
2
−
x
y
∣
|y^2 - xy|
∣
y
2
−
x
y
∣
for
y
y
y
in
[
0
,
1
]
[0, 1]
[
0
,
1
]
. Find the minimum value of
g
g
g
(for real
x
x
x
).
2
1
Hide problems
tan (\pi /3n) is irrational
Show that
tan
π
3
n
\tan \frac{\pi}{3n}
tan
3
n
π
is irrational for all positive integers
n
n
n
.
1
1
Hide problems
m^3 = n^3 + n
Find all integers m, n such that
m
3
=
n
3
+
n
m^3 = n^3 + n
m
3
=
n
3
+
n
.