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Problems
Contests
International Contests
Nordic
1988 Nordic
1988 Nordic
Part of
Nordic
Subcontests
(4)
4
1
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limit of minimum of sum from k=0 to 2n of x_k
Let
m
n
m_n
m
n
be the smallest value of the function
f
n
(
x
)
=
∑
k
=
0
2
n
x
k
{{f}_{n}}\left( x \right)=\sum\limits_{k=0}^{2n}{{{x}^{k}}}
f
n
(
x
)
=
k
=
0
∑
2
n
x
k
Show that
m
n
→
1
2
m_n \to \frac{1}{2}
m
n
→
2
1
, as
n
→
∞
.
n \to \infty.
n
→
∞.
2
1
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(a^3 - c^3)/3 \ge abc [(a- b)/c+ (b- c)/a]
Let
a
,
b
,
a, b,
a
,
b
,
and
c
c
c
be non-zero real numbers and let
a
≥
b
≥
c
a \ge b \ge c
a
≥
b
≥
c
. Prove the inequality
a
3
−
c
3
3
≥
a
b
c
(
a
−
b
c
+
b
−
c
a
)
\frac{a^3 - c^3}{3} \ge abc (\frac{a- b}{c}+ \frac{b- c}{a})
3
a
3
−
c
3
≥
ab
c
(
c
a
−
b
+
a
b
−
c
)
. When does equality hold?
3
1
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2 concentric spheres, triangle ABC of big tangent to small
Two concentric spheres have radii
r
r
r
and
R
,
r
<
R
R,r < R
R
,
r
<
R
. We try to select points
A
,
B
A, B
A
,
B
and
C
C
C
on the surface of the larger sphere such that all sides of the triangle
A
B
C
ABC
A
BC
would be tangent to the surface of the smaller sphere. Show that the points can be selected if and only if
R
≤
2
r
R \le 2r
R
≤
2
r
.
1
1
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three last digits of n and \sqrt[3]{n} remains
The positive integer
n
n
n
has the following property: if the three last digits of
n
n
n
are removed, the number
n
3
\sqrt[3]{n}
3
n
remains. Find
n
n
n
.