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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2000 Swedish Mathematical Competition
2000 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
Hide problems
y(x+y)^2 = 9, y(x^3-y^3) = 7
Solve
{
y
(
x
+
y
)
2
=
9
y
(
x
3
−
y
3
)
=
7
\left\{ \begin{array}{l} y(x+y)^2 = 9 \\ y(x^3-y^3) = 7 \\ \end{array} \right.
{
y
(
x
+
y
)
2
=
9
y
(
x
3
−
y
3
)
=
7
5
1
Hide problems
unique f if f(prime) = 1; f(ab) = a f(b) + f(a) b
Let
f
(
n
)
f(n)
f
(
n
)
be defined on the positive integers and satisfy:
f
(
p
r
i
m
e
)
=
1
f(prime) = 1
f
(
p
r
im
e
)
=
1
,
f
(
a
b
)
=
a
f
(
b
)
+
f
(
a
)
b
f(ab) = a f(b) + f(a) b
f
(
ab
)
=
a
f
(
b
)
+
f
(
a
)
b
. Show that
f
f
f
is unique and find all
n
n
n
such that
n
=
f
(
n
)
n = f(n)
n
=
f
(
n
)
.
4
1
Hide problems
3D lattice points form a triangle with area >= 1/2
The vertices of a triangle are three-dimensional lattice points. Show that its area is at least
1
2
\frac12
2
1
.
3
1
Hide problems
n^2 + (n+1)^2 + (n+2)^2 = m^2
Are there any integral solutions to
n
2
+
(
n
+
1
)
2
+
(
n
+
2
)
2
=
m
2
n^2 + (n+1)^2 + (n+2)^2 = m^2
n
2
+
(
n
+
1
)
2
+
(
n
+
2
)
2
=
m
2
?
2
1
Hide problems
p(y^2-1) =? if p(y^2+1) = 6y^4 - y^2 + 5
p
(
x
)
p(x)
p
(
x
)
is a polynomial such that
p
(
y
2
+
1
)
=
6
y
4
−
y
2
+
5
p(y^2+1) = 6y^4 - y^2 + 5
p
(
y
2
+
1
)
=
6
y
4
−
y
2
+
5
. Find
p
(
y
2
−
1
)
p(y^2-1)
p
(
y
2
−
1
)
.
1
1
Hide problems
each of the numbers 1, 2, ... , 10 is colored red or blue.
Each of the numbers
1
,
2
,
.
.
.
,
10
1, 2, ... , 10
1
,
2
,
...
,
10
is colored red or blue.
5
5
5
is red and at least one number is blue. If
m
,
n
m, n
m
,
n
are different colors and
m
+
n
≤
10
m+n \le 10
m
+
n
≤
10
, then
m
+
n
m+n
m
+
n
is blue. If
m
,
n
m, n
m
,
n
are different colors and
m
n
≤
10
mn \le 10
mn
≤
10
, then
m
n
mn
mn
is red. Find all the colors.