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Problems
Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1988 Swedish Mathematical Competition
1988 Swedish Mathematical Competition
Part of
Swedish Mathematical Competition
Subcontests
(6)
6
1
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inequality when a_{n+1} = \sqrt{a_n^2 + 1/a_n }
The sequence
(
a
n
)
(a_n)
(
a
n
)
is defined by
a
1
=
1
a_1 = 1
a
1
=
1
and
a
n
+
1
=
a
n
2
+
1
a
n
a_{n+1} = \sqrt{a_n^2 +\frac{1}{a_n}}
a
n
+
1
=
a
n
2
+
a
n
1
for
n
≥
1
n \ge 1
n
≥
1
. Prove that there exists
a
a
a
such that
1
2
≤
a
n
n
a
≤
2
\frac{1}{2} \le \frac{a_n}{n^a} \le 2
2
1
≤
n
a
a
n
≤
2
for
n
≥
1
n \ge 1
n
≥
1
.
5
1
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7- m^2/n^2 >= a/n^2 if m/n < \sqrt7
Show that there exists a constant
a
>
1
a > 1
a
>
1
such that, for any positive integers
m
m
m
and
n
n
n
,
m
n
<
7
\frac{m}{n} < \sqrt7
n
m
<
7
implies that
7
−
m
2
n
2
≥
a
n
2
.
7-\frac{m^2}{n^2} \ge \frac{a}{n^2} .
7
−
n
2
m
2
≥
n
2
a
.
4
1
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no fo real roots of P'(x)^2 -2P(x)P''(x) = 0 if P has 3 real distinct
A polynomial
P
(
x
)
P(x)
P
(
x
)
of degree
3
3
3
has three distinct real roots. Find the number of real roots of the equation
P
′
(
x
)
2
−
2
P
(
x
)
P
′
′
(
x
)
=
0
P'(x)^2 -2P(x)P''(x) = 0
P
′
(
x
)
2
−
2
P
(
x
)
P
′′
(
x
)
=
0
.
3
1
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find n such that x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 <= 0 if sum x_i=0
Show that if
x
1
+
x
2
+
x
3
=
0
x_1+x_2+x_3 = 0
x
1
+
x
2
+
x
3
=
0
for real numbers
x
1
,
x
2
,
x
3
x_1,x_2,x_3
x
1
,
x
2
,
x
3
, then
x
1
x
2
+
x
2
x
3
+
x
3
x
1
≤
0
x_1x_2+x_2x_3+x_3x_1\le 0
x
1
x
2
+
x
2
x
3
+
x
3
x
1
≤
0
.Find all
n
≥
4
n \ge 4
n
≥
4
for which
x
1
+
x
2
+
.
.
.
+
x
n
=
0
x_1+x_2+...+x_n = 0
x
1
+
x
2
+
...
+
x
n
=
0
implies
x
1
x
2
+
x
2
x
3
+
.
.
.
+
x
n
−
1
x
n
+
x
n
x
1
≤
0
x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 \le 0
x
1
x
2
+
x
2
x
3
+
...
+
x
n
−
1
x
n
+
x
n
x
1
≤
0
.
2
1
Hide problems
6 ducklings swim on the surface of a cyclic pond
Six ducklings swim on the surface of a pond, which is in the shape of a circle with radius
5
5
5
m. Show that at every moment, two of the ducklings swim on the distance of at most
5
5
5
m from each other.
1
1
Hide problems
a+h_a > b+h_b > c+h_c if a > b > c, and h_i their altitudes
Let
a
>
b
>
c
a > b > c
a
>
b
>
c
be sides of a triangle and
h
a
,
h
b
,
h
c
h_a,h_b,h_c
h
a
,
h
b
,
h
c
be the corresponding altitudes. Prove that
a
+
h
a
>
b
+
h
b
>
c
+
h
c
a+h_a > b+h_b > c+h_c
a
+
h
a
>
b
+
h
b
>
c
+
h
c
.