MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2017 Junior Balkan Team Selection Tests - Moldova
2017 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(8)
Problem 8
1
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Minimum number of moves
The bottom line of a
2
×
13
2\times 13
2
×
13
rectangle is filled with
13
13
13
tokens marked with the numbers
1
,
2
,
.
.
.
,
13
1, 2, ..., 13
1
,
2
,
...
,
13
and located in that order. An operation is a move of a token from its cell into a free adjacent cell (two cells are called adjacent if they have a common side). What is the minimum number of operations needed to rearrange the chips in reverse order in the bottom line of the rectangle?
Problem 2
1
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Inequality
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be the sidelengths of a triangle. Prove that
2
<
a
b
+
c
+
b
c
+
a
+
c
a
+
b
<
6
.
2<\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}<\sqrt{6}.
2
<
b
+
c
a
+
c
+
a
b
+
a
+
b
c
<
6
.
Problem 1
1
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Diophantine equation
Find all natural numbers
x
,
y
x,y
x
,
y
such that
x
5
=
y
5
+
10
y
2
+
20
y
+
1.
x^5=y^5+10y^2+20y+1.
x
5
=
y
5
+
10
y
2
+
20
y
+
1.
Problem 7
1
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Perpendicularity
Given is an acute triangle
A
B
C
ABC
A
BC
and the median
A
M
.
AM.
A
M
.
Draw
B
H
⊥
A
C
.
BH\perp AC.
B
H
⊥
A
C
.
The line which goes through
A
A
A
and is perpendicular to
A
M
AM
A
M
intersects
B
H
BH
B
H
at
E
.
E.
E
.
On the opposite ray of the ray
A
E
AE
A
E
choose
F
F
F
such that
A
E
=
A
F
.
AE=AF.
A
E
=
A
F
.
Prove that
C
F
⊥
A
B
.
CF\perp AB.
CF
⊥
A
B
.
Problem 5
1
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Sequence
Consider the following increasing sequence
1
,
3
,
5
,
7
,
9
,
…
1,3,5,7,9,…
1
,
3
,
5
,
7
,
9
,
…
of all positive integers consisting only of odd digits. Find the
2017
2017
2017
-th term of the above sequence.
Problem 6
1
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Maximum and minimum
Let
a
,
b
a,b
a
,
b
and
c
c
c
be real numbers such that
∣
a
+
b
∣
+
∣
b
+
c
∣
+
∣
c
+
a
∣
=
8.
|a+b|+|b+c|+|c+a|=8.
∣
a
+
b
∣
+
∣
b
+
c
∣
+
∣
c
+
a
∣
=
8.
Find the maximum and minimum value of the expression
P
=
a
2
+
b
2
+
c
2
.
P=a^2+b^2+c^2.
P
=
a
2
+
b
2
+
c
2
.
Problem 3
1
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Line bisects altitude
Let
A
B
C
ABC
A
BC
be a triangle inscribed in a semicircle with center
O
O
O
and diameter
B
C
.
BC.
BC
.
Two tangent lines to the semicircle at
A
A
A
and
B
B
B
intersect at
D
.
D.
D
.
Prove that
D
C
DC
D
C
goes through the midpoint of the altitude
A
H
AH
A
H
of triangle
A
B
C
.
ABC.
A
BC
.
Problem 4
1
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Set with property
Find the maximum positive integer
k
k
k
such that there exist
k
k
k
positive integers which do not exceed
2017
2017
2017
and have the property that every number among them cannot be a power of any of the remaining
k
−
1
k-1
k
−
1
numbers.