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National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2016 Swedish Mathematical Competition
2016 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
5
1
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a new multiplication table for 4 numbers : 1, 2, 3, 4
Peter wants to create a new multiplication table for the four numbers
1
,
2
,
3
,
4
1, 2, 3, 4
1
,
2
,
3
,
4
in such a way that the product of two of them is also one of them. He wants also that
(
a
⋅
b
)
⋅
c
=
a
⋅
(
b
⋅
c
)
(a\cdot b)\cdot c = a\cdot (b\cdot c)
(
a
⋅
b
)
⋅
c
=
a
⋅
(
b
⋅
c
)
holds and that
a
b
≠
a
c
ab \ne ac
ab
=
a
c
and
b
a
≠
c
a
ba \ne ca
ba
=
c
a
and
b
≠
c
b \ne c
b
=
c
. Peter is successful in constructing the new table. In his new table,
1
⋅
3
=
2
1\cdot 3 = 2
1
⋅
3
=
2
and
2
⋅
2
=
4
2\cdot 2 = 4
2
⋅
2
=
4
. What is the product
3
⋅
1
3\cdot 1
3
⋅
1
according to Peter's table?
6
1
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bw in a 13x13 grid
Each cell in a
13
×
13
13 \times 13
13
×
13
grid table is painted in black or white. Each move consists of choosing a subsquare of size either
2
×
2
2 \times 2
2
×
2
or
9
×
9
9 \times 9
9
×
9
, and painting all white cells of the choosen subsquare black, and painting all its black cells white. It is always possible to get all cells of the original square black, after a finite number of such moves ?
4
1
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p + 1 = produsct of primes < p
Find all prime numbers
p
p
p
, for which the number
p
+
1
p + 1
p
+
1
is equal to the product of all the prime numbers which are smaller than
p
p
p
.
2
1
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| \sqrt{x^2+2x+5} - \sqrt{x^2-4x+8}| <3
Determine whether the inequality
∣
x
2
+
2
x
+
5
−
x
2
−
4
x
+
8
∣
<
3
\left|\sqrt{x^2+2x+5}-\sqrt{x^2-4x+8}\right|<3
x
2
+
2
x
+
5
−
x
2
−
4
x
+
8
<
3
is valid for all real numbers
x
x
x
.
1
1
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max area of quadrilateral with sides 20,20,14,13 and right ange between 20s
In a garden there is an
L
L
L
-shaped fence, see figure. You also have at your disposal two finished straight fence sections that are
13
13
13
m and
14
14
14
m long respectively. From point
A
A
A
you want to delimit a part of the garden with an area of at least
200
200
200
m
2
^2
2
. Is it possible to do this? https://1.bp.blogspot.com/-VLWIImY7HBA/X0yZq5BrkTI/AAAAAAAAMbg/8CyP6DzfZTE5iX01Qab3HVrTmaUQ7PvcwCK4BGAYYCw/s400/sweden%2B16p1.png
3
1
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computational with isosceles trapezoid, inscribed, <DAE=90^o, AE /AB=3/4
The quadrilateral
A
B
C
D
ABCD
A
BC
D
is an isosceles trapezoid, where
A
B
∥
C
D
AB\parallel CD
A
B
∥
C
D
. The trapezoid is inscribed in a circle with radius
R
R
R
and center on side
A
B
AB
A
B
. Point
E
E
E
lies on the circumscribed circle and is such that
∠
D
A
E
=
9
0
o
\angle DAE = 90^o
∠
D
A
E
=
9
0
o
. Given that
A
E
A
B
=
3
4
\frac{AE}{AB}=\frac34
A
B
A
E
=
4
3
, calculate the length of the sides of the isosceles trapezoid.