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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2018 Swedish Mathematical Competition
2018 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
3
1
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sequences with patterns of strict inequalities
Let m be a positive integer. An
m
m
m
-pattern is a sequence of
m
m
m
symbols of strict inequalities. An
m
m
m
-pattern is said to be realized by a sequence of
m
+
1
m + 1
m
+
1
real numbers when the numbers meet each of the inequalities in the given order. (For example, the
5
5
5
-pattern
<
,
<
,
>
,
<
,
>
<, <,>, < ,>
<
,
<
,
>
,
<
,
>
is realized by the sequence of numbers
1
,
4
,
7
,
−
3
,
1
,
0
1, 4, 7, -3, 1, 0
1
,
4
,
7
,
−
3
,
1
,
0
.) Given
m
m
m
, which is the least integer
n
n
n
for which there exists any number sequence
x
1
,
.
.
.
,
x
n
x_1,... , x_n
x
1
,
...
,
x
n
such that each
m
m
m
-pattern is realized by a subsequence
x
i
1
,
.
.
.
,
x
i
m
+
1
x_{i_1},... , x_{i_{m + 1}}
x
i
1
,
...
,
x
i
m
+
1
with
1
≤
i
1
<
.
.
.
<
i
m
+
1
≤
n
1 \le i_1 <... < i_{m + 1} \le n
1
≤
i
1
<
...
<
i
m
+
1
≤
n
?
4
1
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1 among n consecutive pos. integers, is divisible by it's sum of digits
Find the least positive integer
n
n
n
with the property: Among arbitrarily
n
n
n
selected consecutive positive integers, all smaller than
2018
2018
2018
, there is at least one that is divisible by its sum of digits .
6
1
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p(x) = 1 + x^n + x^{2n} product of 3 integer polynomial of degree >=1
For which positive integers
n
n
n
can the polynomial
p
(
x
)
=
1
+
x
n
+
x
2
n
p(x) = 1 + x^n + x^{2n}
p
(
x
)
=
1
+
x
n
+
x
2
n
is written as a product of two polynomials with integer coefficients (of degree
≥
1
\ge 1
≥
1
)?
2
1
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f (x) + 2f (\sqrt[3]{1-x^3}) = x^3
Find all functions
f
:
R
→
R
f: R \to R
f
:
R
→
R
that satisfy
f
(
x
)
+
2
f
(
1
−
x
3
3
)
=
x
3
f (x) + 2f (\sqrt[3]{1-x^3}) = x^3
f
(
x
)
+
2
f
(
3
1
−
x
3
)
=
x
3
for all real
x
x
x
. (Here
x
3
\sqrt[3]{x}
3
x
is defined all over
R
R
R
.)
5
1
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min k such that PQ <= k (BP + QC) , angle trisectors related
In a triangle
A
B
C
ABC
A
BC
, two lines are drawn that together trisect the angle at
A
A
A
. These intersect the side
B
C
BC
BC
at points
P
P
P
and
Q
Q
Q
so that
P
P
P
is closer to
B
B
B
and
Q
Q
Q
is closer to
C
C
C
. Determine the smallest constant
k
k
k
such that
∣
P
Q
∣
≤
k
(
∣
B
P
∣
+
∣
Q
C
∣
)
| P Q | \le k (| BP | + | QC |)
∣
PQ
∣
≤
k
(
∣
BP
∣
+
∣
QC
∣
)
, for all such triangles. Determine if there are triangles for which equality applies.
1
1
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parallel angle bisectors to the angle bisectors of diagonals in a cyclic ABCD
Let the
A
B
C
D
ABCD
A
BC
D
be a quadrilateral without parallel sides, inscribed in a circle. Let
P
P
P
and
Q
Q
Q
be the intersection points between the lines containing the quadrilateral opposite sides. Show that the bisectors to the angles at
P
P
P
and
Q
Q
Q
are parallel to the bisectors of the angles at the intersection point of the diagonals of the quadrilateral.