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Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2007 Junior Balkan Team Selection Tests - Moldova
2007 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(8)
8
1
Hide problems
(1+1/2)(1+1/3)(1+1/4)...(1+1/2006)(1+1/2007)
a) Calculate the product
(
1
+
1
2
)
(
1
+
1
3
)
(
1
+
1
4
)
.
.
.
(
1
+
1
2006
)
(
1
+
1
2007
)
\left(1+\frac{1}{2}\right) \left(1+\frac{1}{3}\right) \left(1+\frac{1}{4}\right)... \left(1+\frac{1}{2006}\right) \left(1+\frac{1}{2007}\right)
(
1
+
2
1
)
(
1
+
3
1
)
(
1
+
4
1
)
...
(
1
+
2006
1
)
(
1
+
2007
1
)
b) Let the set
A
=
{
1
2
,
1
3
,
1
4
,
.
.
.
,
1
2006
,
1
2007
}
A =\left\{\frac{1}{2}, \frac{1}{3},\frac{1}{4}, ...,\frac{1}{2006}, \frac{1}{2007}\right\}
A
=
{
2
1
,
3
1
,
4
1
,
...
,
2006
1
,
2007
1
}
Determine the sum of all products of
2
2
2
, of
4
4
4
, of
6
6
6
,... , of
2004
2004
2004
¸and of
2006
2006
2006
different elements of the set
A
A
A
.
7
1
Hide problems
square with side length 14 covered by 21 squares
Show that there is a square with side length
14
14
14
whose floor may be covered (exact coverage of the square area) by
21
21
21
squares so that between them there is exactly
6
6
6
squares with side length
1
1
1
,
5
5
5
squares with side length
2
2
2
,
4
4
4
squares with side length
3
3
3
,
3
3
3
squares with side length
4
4
4
,
2
2
2
squares with side length
5
5
5
and a square with side length
6
6
6
.
6
1
Hide problems
ax^2 + bx + kc = 0, a<b<c, smallest triangle angle <=18^o
The lengths of the sides
a
,
b
a, b
a
,
b
and
c
c
c
of a right triangle satisfy the relations
a
<
b
<
c
a <b <c
a
<
b
<
c
, and
α
\alpha
α
is the measure of the smallest angle of the triangle. For which real values
k
k
k
the equation
a
x
2
+
b
x
+
k
c
=
0
ax^2 + bx + kc = 0
a
x
2
+
b
x
+
k
c
=
0
has real solutions for any measure of the angle
α
\alpha
α
not exceeding
1
8
o
18^o
1
8
o
5
1
Hide problems
min no such that product of digits = 10!
Determine the smallest natural number written in the decimal system with the product of the digits equal to
10
!
=
1
⋅
2
⋅
3
⋅
.
.
.
⋅
9
⋅
10
10! = 1 \cdot 2 \cdot 3\cdot ... \cdot9\cdot10
10
!
=
1
⋅
2
⋅
3
⋅
...
⋅
9
⋅
10
.
2
1
Hide problems
sum 1/(a_1+ a_2) >1 if a_i^2 /(a_i-1)>S, a_>1, sum a_i=S
The real numbers
a
1
,
a
2
,
a
3
a_1, a_2, a_3
a
1
,
a
2
,
a
3
are greater than
1
1
1
and have the sum equal to
S
S
S
. If for any
i
=
1
,
2
,
3
i = 1, 2, 3
i
=
1
,
2
,
3
, holds the inequality
a
i
2
a
i
−
1
>
S
\frac{a_i^2}{a_i-1}>S
a
i
−
1
a
i
2
>
S
, prove the inequality
1
a
1
+
a
2
+
1
a
2
+
a
3
+
1
a
3
+
a
1
>
1
\frac{1}{a_1+ a_2}+\frac{1}{a_2+ a_3}+\frac{1}{a_3+ a_1}>1
a
1
+
a
2
1
+
a
2
+
a
3
1
+
a
3
+
a
1
1
>
1
4
1
Hide problems
average age of the participants in a mathematics competition
The average age of the participants in a mathematics competition (gymnasts and high school students) increases by exactly one month if three high school age students
18
18
18
years each are included in the competition or if three gymnasts aged
12
12
12
years each are excluded from the competition. How many participants were initially in the contest?
1
1
Hide problems
d_1+d_2+...+d_6= 36 iff (d_1-6) (d_2-6) ... (s_6 -6) = -36, digits \ne 6
The numbers
d
1
,
d
2
,
.
.
.
,
d
6
d_1, d_2,..., d_6
d
1
,
d
2
,
...
,
d
6
are distinct digits of the decimal number system other than
6
6
6
. Prove that
d
1
+
d
2
+
.
.
.
+
d
6
=
36
d_1+d_2+...+d_6= 36
d
1
+
d
2
+
...
+
d
6
=
36
if and only if
(
d
1
−
6
)
(
d
2
−
6
)
.
.
.
(
d
6
−
6
)
=
−
36
(d_1-6) (d_2-6) ... (d_6 -6) = -36
(
d
1
−
6
)
(
d
2
−
6
)
...
(
d
6
−
6
)
=
−
36
.
3
1
Hide problems
incenter wanted, 1/AE + 1/AF =( a + b + c)/ bc.
Let
A
B
C
ABC
A
BC
be a triangle with
B
C
=
a
,
A
C
=
b
BC = a, AC = b
BC
=
a
,
A
C
=
b
and
A
B
=
c
AB = c
A
B
=
c
. A point
P
P
P
inside the triangle has the property that for any line passing through
P
P
P
and intersects the lines
A
B
AB
A
B
and
A
C
AC
A
C
in the distinct points
E
E
E
and
F
F
F
we have the relation
1
A
E
+
1
A
F
=
a
+
b
+
c
b
c
\frac{1}{AE} +\frac{1}{AF} =\frac{a + b + c}{bc}
A
E
1
+
A
F
1
=
b
c
a
+
b
+
c
. Prove that the point
P
P
P
is the center of the circle inscribed in the triangle
A
B
C
ABC
A
BC
.