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Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2000 Junior Balkan Team Selection Tests - Moldova
2000 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(8)
4
1
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min n such that x_i \in [1/2 , 2], sum x_i >= 7n/6, sum 1/x_i >= 4n/4
Find the smallest natural number nonzero n so that it exists in real numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2,..., x_n
x
1
,
x
2
,
...
,
x
n
which simultaneously check the conditions: 1)
x
i
∈
[
1
/
2
,
2
]
x_i \in [1/2 , 2]
x
i
∈
[
1/2
,
2
]
,
i
=
1
,
2
,
.
.
.
,
n
i = 1, 2,... , n
i
=
1
,
2
,
...
,
n
2)
x
1
+
x
2
+
.
.
.
+
x
n
≥
7
n
6
x_1+x_2+...+x_n \ge \frac{7n}{6}
x
1
+
x
2
+
...
+
x
n
≥
6
7
n
3)
1
x
1
+
1
x
2
+
.
.
.
+
1
x
n
≥
4
n
3
\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}
x
1
1
+
x
2
1
+
...
+
x
n
1
≥
3
4
n
1
1
Hide problems
(a + b + 1) (a + b - 1) (a - b + 1) (- a + b + 1) constant if a =\sqrt{1 + x^2}
Show that the expression
(
a
+
b
+
1
)
(
a
+
b
−
1
)
(
a
−
b
+
1
)
(
−
a
+
b
+
1
)
(a + b + 1) (a + b - 1) (a - b + 1) (- a + b + 1)
(
a
+
b
+
1
)
(
a
+
b
−
1
)
(
a
−
b
+
1
)
(
−
a
+
b
+
1
)
, where
a
=
1
+
x
2
a =\sqrt{1 + x^2}
a
=
1
+
x
2
,
b
=
1
+
y
2
b =\sqrt{1 + y^2}
b
=
1
+
y
2
and
x
+
y
=
1
x + y = 1
x
+
y
=
1
is constant ¸and be calculated that constant value.
2
1
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lcm of (a_1, a_2, ..., a_{18}) where sum a_1=665
The number
665
665
665
is represented as a sum of
18
18
18
natural numbers nenule
a
1
,
a
2
,
.
.
.
,
a
18
a_1, a_2, ..., a_{18}
a
1
,
a
2
,
...
,
a
18
. Determine the smallest possible value of the smallest common multiple of the numbers
a
1
,
a
2
,
.
.
.
,
a
18
a_1, a_2, ..., a_{18}
a
1
,
a
2
,
...
,
a
18
.
5
1
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sum (a^2-b^2)/c >= 3a - 4b + c if a >= b >= c > 0
Let the real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
be such that
a
≥
b
≥
c
>
0
a \ge b \ge c > 0
a
≥
b
≥
c
>
0
. Show that
a
2
−
b
2
c
+
c
2
−
b
2
a
+
a
2
−
c
2
b
≥
3
a
−
4
b
+
c
.
\frac{a^2-b^2}{c}+ \frac{c^2-b^2}{a}+ \frac{a^2-c^2}{b}\ge 3a - 4b + c.
c
a
2
−
b
2
+
a
c
2
−
b
2
+
b
a
2
−
c
2
≥
3
a
−
4
b
+
c
.
When does equality hold?
7
1
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AA_1, BB_1 , CC_1 are concurrent iff < BAC = 90^o
Let a triangle
A
B
C
,
A
1
ABC, A_1
A
BC
,
A
1
be the midpoint of the segment
[
B
C
]
,
B
1
∈
(
A
C
)
[BC], B_1 \in (AC)
[
BC
]
,
B
1
∈
(
A
C
)
¸and
C
1
∈
(
A
B
)
C_1 \in (AB)
C
1
∈
(
A
B
)
such that
[
A
1
B
1
[A_1B_1
[
A
1
B
1
is the bisector of the angle
A
A
1
C
AA_1C
A
A
1
C
and
A
1
C
1
A_1C_1
A
1
C
1
is perpendicular to
A
B
AB
A
B
. Show that the lines
A
A
1
,
B
B
1
AA_1, BB_1
A
A
1
,
B
B
1
and
C
C
1
CC_1
C
C
1
are concurrent if and only if
∠
B
A
C
=
9
0
o
\angle BAC = 90^o
∠
B
A
C
=
9
0
o
3
1
Hide problems
2AD <BE + EA for angle bisectors in an isosceles 100-40-40 triangle
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
A
C
AB = AC
A
B
=
A
C
¸
∠
B
A
C
=
10
0
o
\angle BAC = 100^o
∠
B
A
C
=
10
0
o
and
A
D
,
B
E
AD, BE
A
D
,
BE
angle bisectors. Prove that
2
A
D
<
B
E
+
E
A
2AD <BE + EA
2
A
D
<
BE
+
E
A
8
1
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Number Theory
Show that the numbers
1
8
n
18^n
1
8
n
and
2
n
+
1
8
n
2^n + 18^n
2
n
+
1
8
n
are having the same number of digits (as written in base 10), for every natural number
n
n
n
.
6
1
Hide problems
Number Theory
Show that among any 39 consecutive natural numbers, there is a number whose sum of the digits is devisible by 11.