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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2013 Swedish Mathematical Competition
2013 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
5
1
Hide problems
2^{n-3}n(n-1) n -tuples of integers fro 3 conditions
Let
n
≥
2
n \geq 2
n
≥
2
be a positive integer. Show that there are exactly
2
n
−
3
n
(
n
−
1
)
2^{n-3}n(n-1)
2
n
−
3
n
(
n
−
1
)
n
n
n
-tuples of integers
(
a
1
,
a
2
,
…
,
a
n
)
(a_1,a_2,\dots,a_n)
(
a
1
,
a
2
,
…
,
a
n
)
, which satisfy the conditions: (i)
a
1
=
0
a_1=0
a
1
=
0
; (ii) for each
m
m
m
,
2
≤
m
≤
n
2 \leq m \leq n
2
≤
m
≤
n
, there is an index in
m
m
m
,
1
≤
i
m
<
m
1 \leq i_m <m
1
≤
i
m
<
m
, such that
∣
a
i
m
−
a
m
∣
≤
1
\left|a_{i_m}-a_m\right|\leq 1
∣
a
i
m
−
a
m
∣
≤
1
; (iii) the
n
n
n
-tuple
(
a
1
,
a
2
,
…
,
a
n
)
(a_1,a_2,\dots,a_n)
(
a
1
,
a
2
,
…
,
a
n
)
contains exactly
n
−
1
n-1
n
−
1
different numbers.
3
1
Hide problems
1 + p^n = m^3
Determine all primes
p
p
p
and all non-negative integers
m
m
m
and
n
n
n
, such that
1
+
p
n
=
m
3
.
1 + p^n = m^3.
1
+
p
n
=
m
3
.
1
1
Hide problems
xy^3z + 2x^3z^3-3x^5y = 0 has odd no of integer solutions in a set
For
r
>
0
r> 0
r
>
0
denote by
B
r
B_r
B
r
the set of points at distance at most
r
r
r
length units from the origin. If
P
r
P_r
P
r
is the set of the points in
B
r
B_r
B
r
whit integer coordinates, show that the equation
x
y
3
z
+
2
x
3
z
3
−
3
x
5
y
=
0
xy^3z + 2x^3z^3-3x^5y = 0
x
y
3
z
+
2
x
3
z
3
−
3
x
5
y
=
0
has an odd number of solutions
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
in
P
r
P_r
P
r
.
6
1
Hide problems
a^2>3b if a^2b^2 + 18 abc > 4b^3+4a^3c+27c^2
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
, be real numbers such that
a
2
b
2
+
18
a
b
c
>
4
b
3
+
4
a
3
c
+
27
c
2
.
a^2b^2 + 18 abc > 4b^3+4a^3c+27c^2 .
a
2
b
2
+
18
ab
c
>
4
b
3
+
4
a
3
c
+
27
c
2
.
Prove that
a
2
>
3
b
a^2>3b
a
2
>
3
b
.
2
1
Hide problems
computational, papefolding a square to get a nonagon
The paper folding art origami is usually performed with square sheets of paper. Someone folds the sheet once along a line through the center of the sheet in orde to get a nonagon. Let
p
p
p
be the perimeter of the nonagon minus the length of the fold, i.e. the total length of the eight sides that are not folds, and denote by s the original side length of the square. Express the area of the nonagon in terms of
p
p
p
and
s
s
s
.
4
1
Hide problems
robotic lawnmower, moves at directions of multiples of 60^o, at most 10 m
A robotic lawnmower is located in the middle of a large lawn. Due a manufacturing defect, the robot can only move straight ahead and turn in directions that are multiples of
6
0
o
60^o
6
0
o
. A fence must be set up so that it delimits the entire part of the lawn that the robot can get to, by traveling along a curve with length no more than
10
10
10
meters from its starting position, given that it is facing north when it starts. How long must the fence be?