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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1965 Swedish Mathematical Competition
1965 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(5)
5
1
Hide problems
f(x) = ax^3 + bx^2 + cx + d, |f(x)| <= 1 for all -1 <= x <= 1
Let
S
S
S
be the set of all real polynomials
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
f(x) = ax^3 + bx^2 + cx + d
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
such that
∣
f
(
x
)
∣
≤
1
|f(x)| \le 1
∣
f
(
x
)
∣
≤
1
for all
−
1
≤
x
≤
1
-1 \le x \le 1
−
1
≤
x
≤
1
. Show that the set of possible
∣
a
∣
|a|
∣
a
∣
for
f
f
f
in
S
S
S
is bounded above and find the smallest upper bound.
4
1
Hide problems
f(1/(1+2x))/f(x) indendent of x if f(x) = \frac{1 + Ax}{1 + Bx}
Find constants
A
>
B
A > B
A
>
B
such that
f
(
1
1
+
2
x
)
f
(
x
)
\frac{f\left( \frac{1}{1+2x}\right) }{f(x)}
f
(
x
)
f
(
1
+
2
x
1
)
is independent of
x
x
x
, where
f
(
x
)
=
1
+
A
x
1
+
B
x
f(x) = \frac{1 + Ax}{1 + Bx}
f
(
x
)
=
1
+
B
x
1
+
A
x
for all real
x
≠
−
1
B
x \ne - \frac{1}{B}
x
=
−
B
1
. Put
a
0
=
1
a_0 = 1
a
0
=
1
,
a
n
+
1
=
1
1
+
2
a
n
a_{n+1} = \frac{1}{1 + 2a_n}
a
n
+
1
=
1
+
2
a
n
1
. Find an expression for an by considering
f
(
a
0
)
,
f
(
a
1
)
,
.
.
.
f(a_0), f(a_1), ...
f
(
a
0
)
,
f
(
a
1
)
,
...
.
3
1
Hide problems
for every x >= 1/2 exists n such that |x - n^2| <= \sqrt{x\frac{1}{4}}.
Show that for every real
x
≥
1
2
x \ge \frac12
x
≥
2
1
there is an integer
n
n
n
such that
∣
x
−
n
2
∣
≤
x
−
1
4
|x - n^2| \le \sqrt{x-\frac{1}{4}}
∣
x
−
n
2
∣
≤
x
−
4
1
.
2
1
Hide problems
m^3 - n^3 = 999
Find all positive integers m, n such that
m
3
−
n
3
=
999
m^3 - n^3 = 999
m
3
−
n
3
=
999
.
1
1
Hide problems
angles of orthic triangles,
The feet of the altitudes in the triangle
A
B
C
ABC
A
BC
are
A
′
,
B
′
,
C
′
A', B', C'
A
′
,
B
′
,
C
′
. Find the angles of
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
in terms of the angles
A
,
B
,
C
A, B, C
A
,
B
,
C
. Show that the largest angle in
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
is at least as big as the largest angle in
A
B
C
ABC
A
BC
. When is it equal?