MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1974 Swedish Mathematical Competition
1974 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
Hide problems
a_1^2+a_2^2+...+a_n^2 a perfect square
For which
n
n
n
can we find positive integers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
such that
a
1
2
+
a
2
2
+
⋯
+
a
n
2
a_1^2+a_2^2+\cdots+a_n^2
a
1
2
+
a
2
2
+
⋯
+
a
n
2
is a square?
5
1
Hide problems
x_1 + x_3 = 2t x_2 , x_2 + x_4 = 2t x_3 , x_3 + x_5=2t x_4
Find the smallest positive real
t
t
t
such that
{
x
1
+
x
3
=
2
t
x
2
x
2
+
x
4
=
2
t
x
3
x
3
+
x
5
=
2
t
x
4
\left\{ \begin{array}{l} x_1 + x_3 = 2t x_2 \\ x_2 + x_4 = 2t x_3 \\ x_3 + x_5=2t x_4 \\ \end{array} \right.
⎩
⎨
⎧
x
1
+
x
3
=
2
t
x
2
x
2
+
x
4
=
2
t
x
3
x
3
+
x
5
=
2
t
x
4
has a solution
x
1
x_1
x
1
,
x
2
x_2
x
2
,
x
3
x_3
x
3
,
x
4
x_4
x
4
,
x
5
x_5
x
5
in non-negative reals, not all zero.
4
1
Hide problems
q (x^2 - 2x ) = q(x-2 )^2
Find all polynomials
p
(
x
)
p(x)
p
(
x
)
such that
p
(
x
2
)
=
p
(
x
)
2
p(x^2) = p(x)^2
p
(
x
2
)
=
p
(
x
)
2
for all
x
x
x
. Hence find all polynomials
q
(
x
)
q(x)
q
(
x
)
such that
q
(
x
2
−
2
x
)
=
q
(
x
−
2
)
2
q\left(x^2 - 2x\right) = q\left(x-2\right)^2
q
(
x
2
−
2
x
)
=
q
(
x
−
2
)
2
3
1
Hide problems
last two digits of a_9=9^{a_8}
Let
a
1
=
1
a_1=1
a
1
=
1
,
a
2
=
2
a
1
a_2=2^{a_1}
a
2
=
2
a
1
,
a
3
=
3
a
2
a_3=3^{a_2}
a
3
=
3
a
2
,
a
4
=
4
a
3
a_4=4^{a_3}
a
4
=
4
a
3
,
…
\dots
…
,
a
9
=
9
a
8
a_9 = 9^{a_8}
a
9
=
9
a
8
. Find the last two digits of
a
9
a_9
a
9
.
2
1
Hide problems
1 -1/k <= n (\sqrt[n]{k}-1 ) <= k - 1
Show that
1
−
1
k
≤
n
(
k
n
−
1
)
≤
k
−
1
1 - \frac{1}{k} \leq n\left(\sqrt[n]{k}-1\right) \leq k - 1
1
−
k
1
≤
n
(
n
k
−
1
)
≤
k
−
1
for all positive integers
n
n
n
and positive reals
k
k
k
.
1
1
Hide problems
b_n = \sum\limits_{r+s \leq n} a_ra_s if a_n = 2^{n-1}
Let
a
n
=
2
n
−
1
a_n = 2^{n-1}
a
n
=
2
n
−
1
for
n
>
0
n > 0
n
>
0
. Let
b
n
=
∑
r
+
s
≤
n
a
r
a
s
b_n = \sum\limits_{r+s \leq n} a_ra_s
b
n
=
r
+
s
≤
n
∑
a
r
a
s
Find
b
n
−
b
n
−
1
b_n-b_{n-1}
b
n
−
b
n
−
1
,
b
n
−
2
b
n
−
1
b_n-2b_{n-1}
b
n
−
2
b
n
−
1
and
b
n
b_n
b
n
.