MathDB
Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1972 Swedish Mathematical Competition
1972 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
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we can find m < n such that a_m <= a_n and b_m <= b_n
a
1
,
a
2
,
a
3
,
…
a_1,a_2,a_3,\dots
a
1
,
a
2
,
a
3
,
…
and
b
1
,
b
2
,
b
3
,
…
b_1,b_2,b_3,\dots
b
1
,
b
2
,
b
3
,
…
are sequences of positive integers. Show that we can find
m
<
n
m < n
m
<
n
such that
a
m
≤
a
n
a_m \leq a_n
a
m
≤
a
n
and
b
m
≤
b
n
b_m \leq b_n
b
m
≤
b
n
.
5
1
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\int \limits_0^1 1/(1+x)^n dx > 1- 1/n
Show that
∫
0
1
1
(
1
+
x
)
n
d
x
>
1
−
1
n
\int\limits_0^1 \frac{1}{(1+x)^n} dx > 1-\frac{1}{n}
0
∫
1
(
1
+
x
)
n
1
d
x
>
1
−
n
1
for all positive integers
n
n
n
.
4
1
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log inequalities
Put
x
=
log
10
2
x = \log_{10} 2
x
=
lo
g
10
2
,
y
=
log
10
3
y = \log_{10} 3
y
=
lo
g
10
3
. Then
15
<
16
15 < 16
15
<
16
implies
1
−
x
+
y
<
4
x
1 - x + y < 4x
1
−
x
+
y
<
4
x
, so
1
+
y
<
5
x
1 + y < 5x
1
+
y
<
5
x
. Derive similar inequalities from
80
<
81
80 < 81
80
<
81
and
243
<
250
243 < 250
243
<
250
. Hence show that
0.47
<
log
10
3
<
0.482.
0.47 < \log_{10} 3 < 0.482.
0.47
<
lo
g
10
3
<
0.482.
3
1
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oven temperature wanted, time related
A steak temperature
5
∘
5^\circ
5
∘
is put into an oven. After
15
15
15
minutes, it has temperature
4
5
∘
45^\circ
4
5
∘
. After another
15
15
15
minutes it has temperature
7
7
∘
77^\circ
7
7
∘
. The oven is at a constant temperature. The steak changes temperature at a rate proportional to the difference between its temperature and that of the oven. Find the oven temperature.
2
1
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rectangular grid of streets has m north-south streets, n east-west streets
A rectangular grid of streets has
m
m
m
north-south streets and
n
n
n
east-west streets. For which
m
,
n
>
1
m, n > 1
m
,
n
>
1
is it possible to start at an intersection and drive through each of the other intersections just once before returning to the start?
1
1
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max a, such that system : x - 4y = 1, ax + 3y = 1 has an integer solution
Find the largest real number
a
a
a
such that
{
x
−
4
y
=
1
a
x
+
3
y
=
1
\left\{ \begin{array}{l} x - 4y = 1 \\ ax + 3y = 1\\ \end{array} \right.
{
x
−
4
y
=
1
a
x
+
3
y
=
1
has an integer solution.