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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2003 Swedish Mathematical Competition
2003 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
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infinite square board with an integer writter in each square, <=N zeroes
Consider an infinite square board with an integer written in each square. Assume that for each square the integer in it is equal to the sum of its neighbor to the left and its neighbor above. Assume also that there exists a row
R
0
R_0
R
0
in the board such that all numbers in
R
0
R_0
R
0
are positive. Denote by
R
1
R_1
R
1
the row below
R
0
R_0
R
0
, by
R
2
R_2
R
2
the row below
R
1
R_1
R
1
etc. Show that for each
N
≥
1
N \ge 1
N
≥
1
the row
R
N
R_N
R
N
cannot contain more than
N
N
N
zeroes.
5
1
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no of quadrilaterals ANCD if AB = a,BC = CD = DA = b, <B = 90^o
Given two positive numbers
a
,
b
a, b
a
,
b
, how many non-congruent plane quadrilaterals are there such that
A
B
=
a
AB = a
A
B
=
a
,
B
C
=
C
D
=
D
A
=
b
BC = CD = DA = b
BC
=
C
D
=
D
A
=
b
and
∠
B
=
9
0
o
\angle B = 90^o
∠
B
=
9
0
o
?
4
1
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1 + P(x) = 1/2 (P(x -1) + P(x + 1))
Determine all polynomials
P
P
P
with real coeffients such that
1
+
P
(
x
)
=
1
2
(
P
(
x
−
1
)
+
P
(
x
+
1
)
)
1 + P(x) = \frac12 (P(x -1) + P(x + 1))
1
+
P
(
x
)
=
2
1
(
P
(
x
−
1
)
+
P
(
x
+
1
))
for all real
x
x
x
.
3
1
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[x^2 - 2x] + 2[x] = [x]^2
Find all real solutions
x
x
x
of the equation
⌊
x
2
−
2
⌋
+
2
⌊
x
⌋
=
⌊
x
⌋
2
.
\lfloor x^2-2 \rfloor +2 \lfloor x \rfloor = \lfloor x \rfloor ^2.
⌊
x
2
−
2
⌋
+
2
⌊
x
⌋
=
⌊
x
⌋
2
.
.
2
1
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6 boys sit in each row, 8 girls sit in in each column
In a lecture hall some chairs are placed in rows and columns, forming a rectangle. In each row there are
6
6
6
boys sitting and in each column there are
8
8
8
girls sitting, whereas
15
15
15
places are not taken. What can be said about the number of rows and that of columns?
1
1
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min x if y = x - 2003, z = 2y - 2003, w = 3z - 2003 ofr x,y,z,ww>=0
If
x
,
y
,
z
,
w
x, y, z, w
x
,
y
,
z
,
w
are nonnegative real numbers satisfying
{
y
=
x
−
2003
z
=
2
y
−
2003
w
=
3
z
−
2003
\left\{ \begin{array}{l}y = x - 2003 \\ z = 2y - 2003 \\ w = 3z - 2003 \\ \end{array} \right.
⎩
⎨
⎧
y
=
x
−
2003
z
=
2
y
−
2003
w
=
3
z
−
2003
find the smallest possible value of
x
x
x
and the values of
y
,
z
,
w
y, z, w
y
,
z
,
w
corresponding to it.