MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1963 Swedish Mathematical Competition.
1963 Swedish Mathematical Competition.
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
Hide problems
|f'(x)| ≤ C if |f(x)| ≤ A, |f''(x)| ≤ B
The real-valued function
f
(
x
)
f(x)
f
(
x
)
is defined on the reals. It satisfies
∣
f
(
x
)
∣
≤
A
|f(x)| \le A
∣
f
(
x
)
∣
≤
A
,
∣
f
′
′
(
x
)
∣
≤
B
|f''(x)| \le B
∣
f
′′
(
x
)
∣
≤
B
for some positive
A
,
B
A, B
A
,
B
(and all
x
x
x
). Show that
∣
f
′
(
x
)
∣
≤
C
|f'(x)| \le C
∣
f
′
(
x
)
∣
≤
C
, for some fixed
C
C
C
, which depends only on
A
A
A
and
B
B
B
. What is the smallest possible value of
C
C
C
?
5
1
Hide problems
shortest route from A to B
A road has constant width. It is made up of finitely many straight segments joined by corners, where the inner corner is a point and the outer side is a circular arc. The direction of the straight sections is always between
N
E
NE
NE
(
4
5
o
45^o
4
5
o
) and
S
S
E
SSE
SSE
(
1571
/
2
o
157 1/2^o
1571/
2
o
). A person wishes to walk along the side of the road from point
A
A
A
to point
B
B
B
on the same side. He may only cross the street perpendicularly. What is the shortest route?[figure missing]
4
1
Hide problems
f(x+y) = f(x) + f(y) + f(kxy) , differentiable
Given the real number
k
k
k
, find all differentiable real-valued functions
f
(
x
)
f(x)
f
(
x
)
defined on the reals such that
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
f
(
k
x
y
)
f(x+y) = f(x) + f(y) + f(kxy)
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
+
f
(
k
x
y
)
for all
x
,
y
x, y
x
,
y
.
3
1
Hide problems
remainder of 1234^{567} + 89^{1011} when divided by 12
What is the remainder on dividing
123
4
567
+
8
9
1011
1234^{567} + 89^{1011}
123
4
567
+
8
9
1011
by
12
12
12
?
2
1
Hide problems
max circle drawn entirely on black squares of the chessboard
The squares of a chessboard have side
4
4
4
. What is the circumference of the largest circle that can be drawn entirely on the black squares of the board?
1
1
Hide problems
no of positive integers with square less than 10^7
How many positive integers have square less than
1
0
7
10^7
1
0
7
?