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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2011 Swedish Mathematical Competition
2011 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
5
1
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2 player game in 11x11 grid
Arne and Bertil play a game on an
11
×
11
11 \times 11
11
×
11
grid. Arne starts. He has a game piece that is placed on the center od the grid at the beginning of the game. At each move he moves the piece one step horizontally or vertically. Bertil places a wall along each move any of an optional four squares. Arne is not allowed to move his piece through a wall. Arne wins if he manages to move the pice out of the board, while Bertil wins if he manages to prevent Arne from doing that. Who wins if from the beginning there are no walls on the game board and both players play optimally?
6
1
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no of functions f (max {x + y + 2, xy} ) = min {f (x + y), f (xy + 2)}
How many functions
f
:
N
→
N
f:\mathbb N \to \mathbb N
f
:
N
→
N
are there such that
f
(
0
)
=
2011
f(0)=2011
f
(
0
)
=
2011
,
f
(
1
)
=
111
f(1) = 111
f
(
1
)
=
111
, and
f
(
max
{
x
+
y
+
2
,
x
y
}
)
=
min
{
f
(
x
+
y
)
,
f
(
x
y
+
2
)
}
f\left(\max \{x + y + 2, xy\}\right) = \min \{f (x + y), f (xy + 2)\}
f
(
max
{
x
+
y
+
2
,
x
y
}
)
=
min
{
f
(
x
+
y
)
,
f
(
x
y
+
2
)}
for all non-negative integers
x
x
x
,
y
y
y
?
4
1
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43 lines between A and B, 29 lines between B and C
Towns
A
A
A
,
B
B
B
and
C
C
C
are connected with a telecommunications cable. If you for example want to send a message from
A
A
A
to
B
B
B
is assigned to either a direct line between
A
A
A
and
B
B
B
, or if necessary, a line via
C
C
C
. There are
43
43
43
lines between
A
A
A
and
B
B
B
, including those who go through
C
C
C
, and
29
29
29
lines between
B
B
B
and
C
C
C
, including those who go via
A
A
A
. How many lines, are there between
A
A
A
and
C
C
C
(including those who go via
B
B
B
)?
3
1
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x - 1/y^2 = y - 1/z^2 = z - /x^2
Find all positive real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
, such that
x
−
1
y
2
=
y
−
1
z
2
=
z
−
1
x
2
x - \frac{1}{y^2} = y - \frac{1}{z^2}= z - \frac{1}{x^2}
x
−
y
2
1
=
y
−
z
2
1
=
z
−
x
2
1
1
1
Hide problems
1 /k ! + 1 /l ! + 1 /m ! = 1 / n !
Determine all positive integers
k
k
k
,
ℓ
\ell
ℓ
,
m
m
m
and
n
n
n
, such that
1
k
!
+
1
ℓ
!
+
1
m
!
=
1
n
!
\frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!}
k
!
1
+
ℓ
!
1
+
m
!
1
=
n
!
1
2
1
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<ABC <90^o when | BP | > | AP |, | BP | > | CP |, P interior of ABC
Given a triangle
A
B
C
ABC
A
BC
, let
P
P
P
be a point inside the triangle such that
∣
B
P
∣
>
∣
A
P
∣
,
∣
B
P
∣
>
∣
C
P
∣
| BP | > | AP |, | BP | > | CP |
∣
BP
∣
>
∣
A
P
∣
,
∣
BP
∣
>
∣
CP
∣
. Show that
∠
A
B
C
<
9
0
o
\angle ABC <90^o
∠
A
BC
<
9
0
o