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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2008 Swedish Mathematical Competition
2008 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
5
1
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Anna and Orjan play a game, writine a pos. integer as sum of 2 others
Anna and Orjan play the following game: they start with a positive integer
n
>
1
n>1
n
>
1
, Anna writes it as the sum of two other positive integers,
n
=
n
1
+
n
2
n = n_1+n_2
n
=
n
1
+
n
2
. Orjan deletes one of them,
n
1
n_1
n
1
or
n
2
n_2
n
2
. If the remaining number is larger than
1
1
1
, the process is repeated, i.e. Anna writes it as the sum of two positive integers,
n
3
+
n
4
n_3+n_4
n
3
+
n
4
, Orjan deletes one of them etc. The game ends when the last number is
1
1
1
. Orjan is the winner if there are two equal numbers among the numbers he has deleted, otherwise Anna wins. Who is winning the game if n = 2008 and they both play optimally?
6
1
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max product of x_i when x_ ordered and sum x_1=100
A sum decomposition of the number 100 is given by a positive integer
n
n
n
and
n
n
n
positive integers
x
1
<
x
2
<
⋯
<
x
n
x_1<x_2<\cdots <x_n
x
1
<
x
2
<
⋯
<
x
n
such that
x
1
+
x
2
+
⋯
+
x
n
=
100
x_1 + x_2 + \cdots + x_n = 100
x
1
+
x
2
+
⋯
+
x
n
=
100
. Determine the largest possible value of the product
x
1
x
2
⋯
x
n
x_1x_2\cdots x_n
x
1
x
2
⋯
x
n
, and
n
n
n
, as
x
1
,
x
2
,
…
,
x
n
x_1, x_2,\dots, x_n
x
1
,
x
2
,
…
,
x
n
vary among all sum decompositions of the number
100
100
100
.
2
1
Hide problems
2 number from 1-n such that their product = sum of other n-2
Determine the smallest integer
n
≥
3
n \ge 3
n
≥
3
with the property that you can choose two of the numbers
1
,
2
,
…
,
n
1,2,\dots, n
1
,
2
,
…
,
n
in such a way that their product is equal to the sum of the other
n
−
2
n - 2
n
−
2
languages. What are the two numbers?
4
1
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integer angles in convex polygon , whose degrees are divisible by 36
A convex
n
n
n
-side polygon has angles
v
1
,
v
2
,
…
,
v
n
v_1,v_2,\dots,v_n
v
1
,
v
2
,
…
,
v
n
(in degrees), where all
v
k
v_k
v
k
(
k
=
1
,
2
,
…
,
n
k = 1,2,\dots,n
k
=
1
,
2
,
…
,
n
) are positive integers divisible by
36
36
36
. (a) Determine the largest
n
n
n
for which this is possible. (b) Show that if
n
>
5
n>5
n
>
5
, two of the sides of the
n
n
n
-polygon must be parallel.
3
1
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f(x)+f(y) <=f(x+y) if f(x)/x is increasing for x>0
The function
f
(
x
)
f(x)
f
(
x
)
has the property that
f
(
x
)
x
\frac{f(x)}{x}
x
f
(
x
)
is increasing for
x
>
0
x>0
x
>
0
. Show that
f
(
x
)
+
f
(
y
)
≤
f
(
x
+
y
)
,
for all
x
,
y
>
0
f(x)+f(y) \leq f(x+y) \qquad , \qquad \text{for all } x,y>0
f
(
x
)
+
f
(
y
)
≤
f
(
x
+
y
)
,
for all
x
,
y
>
0
1
1
Hide problems
computational with a rhombus inscribed in convex quadrilateral
A rhombus is inscribed in a convex quadrilateral. The sides of the rhombus are parallel with the diagonals of the quadrilateral, which have the lengths
d
1
d_1
d
1
and
d
2
d_2
d
2
. Calculate the length of side of the rhombus , expressed in terms of
d
1
d_1
d
1
and
d
2
d_2
d
2
.