MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1997 Swedish Mathematical Competition
1997 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
1
1
Hide problems
1/(a^2+1/2 )< b/a < 1/a^2 , geo inequality related to a circle
Let
A
C
AC
A
C
be a diameter of a circle and
A
B
AB
A
B
be tangent to the circle. The segment
B
C
BC
BC
intersects the circle again at
D
D
D
. Show that if
A
C
=
1
AC = 1
A
C
=
1
,
A
B
=
a
AB = a
A
B
=
a
, and
C
D
=
b
CD = b
C
D
=
b
, then
1
a
2
+
1
2
<
b
a
<
1
a
2
\frac{1}{a^2+ \frac12 }< \frac{b}{a}< \frac{1}{a^2}
a
2
+
2
1
1
<
a
b
<
a
2
1
5
1
Hide problems
s(k)+s(f(n)-k) = n, sum of digits
Let
s
(
m
)
s(m)
s
(
m
)
denote the sum of (decimal) digits of a positive integer
m
m
m
. Prove that for every integer
n
>
1
n > 1
n
>
1
not equal to
10
10
10
there is a unique integer
f
(
n
)
≥
2
f(n) \ge 2
f
(
n
)
≥
2
such that
s
(
k
)
+
s
(
f
(
n
)
−
k
)
=
n
s(k)+s(f(n)-k) = n
s
(
k
)
+
s
(
f
(
n
)
−
k
)
=
n
for all integers
k
k
k
with
0
<
k
<
f
(
n
)
0 < k < f(n)
0
<
k
<
f
(
n
)
.
6
1
Hide problems
M contains a pair of distinct numbers whose difference is an integer
Assume that a set
M
M
M
of real numbers is the union of finitely many disjoint intervals with the total length greater than
1
1
1
. Prove that
M
M
M
contains a pair of distinct numbers whose difference is an integer.
4
1
Hide problems
2 player dice game, create 2-digits numbers, probability wanted
Players
A
A
A
and
B
B
B
play the following game. Each of them throws a dice, and if the outcomes are
x
x
x
and
y
y
y
respectively, a list of all two digit numbers
10
a
+
b
10a + b
10
a
+
b
with
a
,
b
∈
{
1
,
.
.
,
6
}
a,b\in \{1,..,6\}
a
,
b
∈
{
1
,
..
,
6
}
and
10
a
+
b
≤
10
x
+
y
10a + b \le 10x + y
10
a
+
b
≤
10
x
+
y
is created. Then the players alternately reduce the list by replacing a pair of numbers in the list by their absolute difference, until only one number remains. If the remaining number is of the same parity as the outcome of
A
A
A
’s throw, then
A
A
A
is proclaimed the winner. What is the probability that
A
A
A
wins the game?
3
1
Hide problems
every integer can be written in the form x^2 -y^2 +Ax+By where A+B= odd
Let
A
A
A
and
B
B
B
be integers with an odd sum. Show that every integer can be written in the form
x
2
−
y
2
+
A
x
+
B
y
x^2 -y^2 +Ax+By
x
2
−
y
2
+
A
x
+
B
y
, where
x
,
y
x,y
x
,
y
are integers.
2
1
Hide problems
angles wanted, 3<ACE = 2<BCE , ED = DC = CP
Let
D
D
D
be the point on side
A
C
AC
A
C
of a triangle
A
B
C
ABC
A
BC
such that
B
D
BD
B
D
bisects
∠
B
\angle B
∠
B
, and
E
E
E
be the point on side
A
B
AB
A
B
such that
3
∠
A
C
E
=
2
∠
B
C
E
3\angle ACE = 2\angle BCE
3∠
A
CE
=
2∠
BCE
. Suppose that
B
D
BD
B
D
and
C
E
CE
CE
intersect at a point
P
P
P
with
E
D
=
D
C
=
C
P
ED = DC = CP
E
D
=
D
C
=
CP
. Determine the angles of the triangle.