MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1984 Swedish Mathematical Competition
1984 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
3
1
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((a+1)/(b+1))^(b+1) >=(a/b)^b for a,b>0
Prove that if
a
,
b
a,b
a
,
b
are positive numbers, then
(
a
+
1
b
+
1
)
b
+
1
≥
(
a
b
)
b
\left( \frac{a+1}{b+1}\right)^{b+1} \ge \left( \frac{a}{b}\right)^{b}
(
b
+
1
a
+
1
)
b
+
1
≥
(
b
a
)
b
6
1
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a_i, i=1,...,7 consist of the numbers 1,...,7 ifsum 3^{a_i} = 6558
Assume
a
1
,
a
2
,
.
.
.
,
a
14
a_1,a_2,...,a_{14}
a
1
,
a
2
,
...
,
a
14
are positive integers such that
∑
i
=
1
14
3
a
i
=
6558
\sum_{i=1}^{14}3^{a_i} = 6558
∑
i
=
1
14
3
a
i
=
6558
. Prove that the numbers
a
1
,
a
2
,
.
.
.
,
a
14
a_1,a_2,...,a_{14}
a
1
,
a
2
,
...
,
a
14
consist of the numbers
1
,
.
.
.
,
7
1,...,7
1
,
...
,
7
, each taken twice.
5
1
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diophantine, a^3 -b^3 -c^3 = 3abc, a^2 = 2(a+b+c),
Solve in natural numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
the system
{
a
3
−
b
3
−
c
3
=
3
a
b
c
a
2
=
2
(
a
+
b
+
c
)
\left\{ \begin{array}{l}a^3 -b^3 -c^3 = 3abc \\ a^2 = 2(a+b+c)\\ \end{array} \right.
{
a
3
−
b
3
−
c
3
=
3
ab
c
a
2
=
2
(
a
+
b
+
c
)
4
1
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(x^2 - px+q)(x^2 -qx+ p) has pos. integer roots
Find all positive integers
p
p
p
and
q
q
q
such that all the roots of the polynomial
(
x
2
−
p
x
+
q
)
(
x
2
−
q
x
+
p
)
(x^2 - px+q)(x^2 -qx+ p)
(
x
2
−
p
x
+
q
)
(
x
2
−
q
x
+
p
)
are positive integers.
2
1
Hide problems
mxn rectangles in a 3x7 colored blue or yellow grid
The squares in a
3
×
7
3\times 7
3
×
7
grid are colored either blue or yellow. Consider all
m
×
n
m\times n
m
×
n
rectangles in this grid, where
m
∈
{
2
,
3
}
m \in \{2,3\}
m
∈
{
2
,
3
}
,
n
∈
{
2
,
.
.
.
,
7
}
n \in \{2,...,7\}
n
∈
{
2
,
...
,
7
}
. Prove that at least one of these rectangles has all four corner squares the same color.
1
1
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exists circle inside circle that contains 2 of the larger interior points
Let
A
A
A
and
B
B
B
be two points inside a circle
C
C
C
. Show that there exists a circle that contains
A
A
A
and
B
B
B
and lies completely inside
C
C
C
.