MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1971 Swedish Mathematical Competition
1971 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
Hide problems
99 cards each have a label chosen from 1,2,...,99
99
99
99
cards each have a label chosen from
1
,
2
,
…
,
99
1,2,\dots,99
1
,
2
,
…
,
99
, such that no (non-empty) subset of the cards has labels with total divisible by
100
100
100
. Show that the labels must all be equal.
5
1
Hide problems
max |1 - a \cos x| >= \tan^2 (t/2) when |x|<= t, a>0, 0<t< \pi/2
Show that
max
∣
x
∣
≤
t
∣
1
−
a
cos
x
∣
≥
tan
2
t
2
\max\limits_{|x|\leq t} |1 - a \cos x| \geq \tan^2 \frac{t}{2}
∣
x
∣
≤
t
max
∣1
−
a
cos
x
∣
≥
tan
2
2
t
for
a
a
a
positive and
t
∈
(
0
,
π
2
)
t \in (0, \frac{\pi}{2})
t
∈
(
0
,
2
π
)
.
4
1
Hide problems
(65533^3 +...65539^3)/(32765 x 32766 + ..+32770 x 32771)
Find
6553
3
3
+
6553
4
3
+
6553
5
3
+
6553
6
3
+
6553
7
3
+
6553
8
3
+
6553
9
3
32765
⋅
32766
+
32767
⋅
32768
+
32768
⋅
32769
+
32770
⋅
32771
\frac{65533^3 + 65534^3 + 65535^3 + 65536^3 + 65537^3 + 65538^3+ 65539^3}{32765\cdot 32766 + 32767\cdot 32768 + 32768\cdot 32769 + 32770\cdot 32771}
32765
⋅
32766
+
32767
⋅
32768
+
32768
⋅
32769
+
32770
⋅
32771
6553
3
3
+
6553
4
3
+
6553
5
3
+
6553
6
3
+
6553
7
3
+
6553
8
3
+
6553
9
3
3
1
Hide problems
remove 7 from 15 pieces of paper which cover a paper
A table is covered by
15
15
15
pieces of paper. Show that we can remove
7
7
7
pieces so that the remaining
8
8
8
cover at least
8
/
15
8/15
8/15
of the table.
2
1
Hide problems
lines divide the plane into regions, red and blue
An arbitrary number of lines divide the plane into regions. Show that the regions can be colored red and blue so that neighboring regions have different colors.
1
1
Hide problems
(1 + a + a^2 )^2 < 3 (1 + a^2 + a^4 )
Show that
(
1
+
a
+
a
2
)
2
<
3
(
1
+
a
2
+
a
4
)
\left(1 + a + a^2\right)^2 < 3\left(1 + a^2 + a^4\right)
(
1
+
a
+
a
2
)
2
<
3
(
1
+
a
2
+
a
4
)
for real
a
≠
1
a \neq 1
a
=
1
.