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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1993 Swedish Mathematical Competition
1993 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
2
1
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distance between railway stations
A railway line is divided into ten sections by the stations
A
,
B
,
C
,
D
,
E
,
F
,
G
,
H
,
I
,
J
,
K
A,B,C,D,E,F, G,H,I,J,K
A
,
B
,
C
,
D
,
E
,
F
,
G
,
H
,
I
,
J
,
K
. The length of each section is an integer number of kilometers and the distacne between
A
A
A
and
K
K
K
is
56
56
56
km. A trip along two successive sections never exceeds
12
12
12
km, but a trip along three successive sections is at least
17
17
17
km. What is the distance between
B
B
B
and
G
G
G
? https://cdn.artofproblemsolving.com/attachments/1/f/202ddf633ed6da8692bf4d0b1fc0af59548526.png
6
1
Hide problems
f(x_1) = x_2, f(x_2) = x_3 , f(x_3) = x_1 if f(x) = 1/(ax+b)
For real numbers
a
a
a
and
b
b
b
define
f
(
x
)
=
1
a
x
+
b
f(x) = \frac{1}{ax+b}
f
(
x
)
=
a
x
+
b
1
. For which
a
a
a
and
b
b
b
are there three distinct real numbers
x
1
,
x
2
,
x
3
x_1,x_2,x_3
x
1
,
x
2
,
x
3
such that
f
(
x
1
)
=
x
2
f(x_1) = x_2
f
(
x
1
)
=
x
2
,
f
(
x
2
)
=
x
3
f(x_2) = x_3
f
(
x
2
)
=
x
3
and
f
(
x
3
)
=
x
1
f(x_3) = x_1
f
(
x
3
)
=
x
1
?
5
1
Hide problems
sequence of triangles, 1st with sides a,b,c, 2nd has sides p-a,p-b,p-c
A triangle with sides
a
,
b
,
c
a,b,c
a
,
b
,
c
and perimeter
2
p
2p
2
p
is given. Is possible, a new triangle with sides
p
−
a
p-a
p
−
a
,
p
−
b
p-b
p
−
b
,
p
−
c
p-c
p
−
c
is formed. The process is then repeated with the new triangle. For which original triangles can this process be repeated indefinitely?
4
1
Hide problems
sovle x*36 = 216 if a*(b*c) = (a*b)c$ and $a*a = 1$
To each pair of nonzero real numbers
a
a
a
and
b
b
b
a real number
a
∗
b
a*b
a
∗
b
is assigned so that
a
∗
(
b
∗
c
)
=
(
a
∗
b
)
c
a*(b*c) = (a*b)c
a
∗
(
b
∗
c
)
=
(
a
∗
b
)
c
and
a
∗
a
=
1
a*a = 1
a
∗
a
=
1
for all
a
,
b
,
c
a,b,c
a
,
b
,
c
. Solve the equation
x
∗
36
=
216
x*36 = 216
x
∗
36
=
216
.
3
1
Hide problems
a^2 +b^2 +x^2 = y^2 has an integer solution x,y iff product ab is even
Assume that
a
a
a
and
b
b
b
are integers. Prove that the equation
a
2
+
b
2
+
x
2
=
y
2
a^2 +b^2 +x^2 = y^2
a
2
+
b
2
+
x
2
=
y
2
has an integer solution
x
,
y
x,y
x
,
y
if and only if the product
a
b
ab
ab
is even.
1
1
Hide problems
x is divisible by 9 if sums of the digits of x and of 3x are the same
An integer
x
x
x
has the property that the sums of the digits of
x
x
x
and of
3
x
3x
3
x
are the same. Prove that
x
x
x
is divisible by
9
9
9
.