MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1973 Swedish Mathematical Competition
1973 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
Hide problems
f(x) < 12f (x - 1/2)+ 1/2 f (x +1/2)
f
(
x
)
f(x)
f
(
x
)
is a real valued function defined for
x
≥
0
x \geq 0
x
≥
0
such that
f
(
0
)
=
0
f(0) = 0
f
(
0
)
=
0
,
f
(
x
+
1
)
=
f
(
x
)
+
x
f(x+1)=f(x)+\sqrt{x}
f
(
x
+
1
)
=
f
(
x
)
+
x
for all
x
x
x
, and f(x) < \frac{1}{2}f\left(x - \frac{1}{2}\right)+\frac{1}{2}f\left(x + \frac{1}{2}\right) \text{for all} x \geq \frac{1}{2} Show that
f
(
1
2
)
f\left(\frac{1}{2}\right)
f
(
2
1
)
is uniquely determined.
5
1
Hide problems
f(x)q(x)-p(x) of form \sum\limits_{2n<k\leq 3n} (a^k + x^k) have same p(x)/q(x)
f
(
x
)
f(x)
f
(
x
)
is a polynomial of degree
2
n
2n
2
n
. Show that all polynomials
p
(
x
)
p(x)
p
(
x
)
,
q
(
x
)
q(x)
q
(
x
)
of degree at most
n
n
n
such that
f
(
x
)
q
(
x
)
−
p
(
x
)
f(x)q(x)-p(x)
f
(
x
)
q
(
x
)
−
p
(
x
)
has the form
∑
2
n
<
k
≤
3
n
(
a
k
+
x
k
)
\sum\limits_{2n<k\leq 3n} (a^k + x^k)
2
n
<
k
≤
3
n
∑
(
a
k
+
x
k
)
have the same
p
(
x
)
/
q
(
x
)
p(x)/q(x)
p
(
x
)
/
q
(
x
)
.
4
1
Hide problems
m/n + 1 /p^2 =(m+p)/(n+p)
p
p
p
is a prime. Find all relatively prime positive integers
m
m
m
,
n
n
n
such that
m
n
+
1
p
2
=
m
+
p
n
+
p
\frac{m}{n}+\frac{1}{p^2}=\frac{m+p}{n+p}
n
m
+
p
2
1
=
n
+
p
m
+
p
3
1
Hide problems
ratio PA_1:PB_1:PC_1 wanted, starting with a 90-60-30 triangle, equilateral
A
B
C
ABC
A
BC
is a triangle with
∠
A
=
9
0
∘
\angle A = 90^\circ
∠
A
=
9
0
∘
,
∠
B
=
6
0
∘
\angle B = 60^\circ
∠
B
=
6
0
∘
. The points
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
on
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
respectively are such that
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
is equilateral and the perpendiculars (to
B
C
BC
BC
at
A
1
A_1
A
1
, to
C
A
CA
C
A
at
B
1
B_1
B
1
and to
A
B
AB
A
B
at
C
1
C_1
C
1
) meet at a point
P
P
P
inside the triangle. Find the ratios
P
A
1
:
P
B
1
:
P
C
1
PA_1:PB_1:PC_1
P
A
1
:
P
B
1
:
P
C
1
.
2
1
Hide problems
f_n = n^2, Fibonacci
The Fibonacci sequence
f
1
,
f
2
,
f
3
,
…
f_1,f_2,f_3,\dots
f
1
,
f
2
,
f
3
,
…
is defined by
f
1
=
f
2
=
1
f_1=f_2=1
f
1
=
f
2
=
1
,
f
n
+
2
=
f
n
+
1
+
f
n
f_{n+2}=f_{n+1}+f_n
f
n
+
2
=
f
n
+
1
+
f
n
. Find all
n
n
n
such that
f
n
=
n
2
f_n = n^2
f
n
=
n
2
.
1
1
Hide problems
\log_8 4 in base 8 (to 4 places of decimals).
log
8
2
=
0.2525
\log_8 2 = 0.2525
lo
g
8
2
=
0.2525
in base
8
8
8
(to
4
4
4
places of decimals). Find
log
8
4
\log_8 4
lo
g
8
4
in base
8
8
8
(to
4
4
4
places of decimals).