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International Contests
Balkan MO
1998 Balkan MO
1998 Balkan MO
Part of
Balkan MO
Subcontests
(4)
3
1
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Let $\mathcal S$ denote the set of points inside or on the b
Let
S
\mathcal S
S
denote the set of points inside or on the border of a triangle
A
B
C
ABC
A
BC
, without a fixed point
T
T
T
inside the triangle. Show that
S
\mathcal S
S
can be partitioned into disjoint closed segemnts. Yugoslavia
2
1
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Plus minus inequality by enescu
Let
n
≥
2
n\geq 2
n
≥
2
be an integer, and let
0
<
a
1
<
a
2
<
⋯
<
a
2
n
+
1
0 < a_1 < a_2 < \cdots < a_{2n+1}
0
<
a
1
<
a
2
<
⋯
<
a
2
n
+
1
be real numbers. Prove the inequality
a
1
n
−
a
2
n
+
a
3
n
−
⋯
+
a
2
n
+
1
n
<
a
1
−
a
2
+
a
3
−
⋯
+
a
2
n
+
1
n
.
\sqrt[n]{a_1} - \sqrt[n]{a_2} + \sqrt[n]{a_3} - \cdots + \sqrt[n]{a_{2n+1}} < \sqrt[n]{a_1 - a_2 + a_3 - \cdots + a_{2n+1}}.
n
a
1
−
n
a
2
+
n
a
3
−
⋯
+
n
a
2
n
+
1
<
n
a
1
−
a
2
+
a
3
−
⋯
+
a
2
n
+
1
.
Bogdan Enescu, Romania
1
1
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Finite sequence - how many distinct terms?
Consider the finite sequence
⌊
k
2
1998
⌋
\left\lfloor \frac{k^2}{1998} \right\rfloor
⌊
1998
k
2
⌋
, for
k
=
1
,
2
,
…
,
1997
k=1,2,\ldots, 1997
k
=
1
,
2
,
…
,
1997
. How many distinct terms are there in this sequence? Greece
4
1
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x^2+4 = y^5
Prove that the following equation has no solution in integer numbers:
x
2
+
4
=
y
5
.
x^2 + 4 = y^5.
x
2
+
4
=
y
5
.
Bulgaria