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Problems
Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2022 Junior Balkan Team Selection Tests - Moldova
2022 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(12)
12
1
Hide problems
square trinomial with integer roots
Let
p
p
p
and
q
q
q
be two distinct integers. The square trinomial
x
2
+
p
x
+
q
x^2 + px + q
x
2
+
p
x
+
q
is written on the board. At each step, a number is deleted: or the coefficient next to
x
x
x
, or the free term, and instead of the deleted number, a number is written, which is obtained from the deleted number by adding or subtracting the number
1
1
1
. After several such steps on the board, the square trinomial
x
2
+
q
x
+
p
x^2 + qx + p
x
2
+
q
x
+
p
appeared. Show that at one stage a square trinomial was written on the board, both roots of which are integers.
11
1
Hide problems
2m | (3n - 2) and 2n | (3m- 2)
Find all ordered pairs of positive integers
(
m
,
n
)
(m, n)
(
m
,
n
)
such that
2
m
2m
2
m
divides the number
3
n
−
2
3n - 2
3
n
−
2
, and
2
n
2n
2
n
divides the number
3
m
−
2
3m - 2
3
m
−
2
.
10
1
Hide problems
2 [x] {x} = x^2 - 3/2 x - 11/16
Solve in the set
R
R
R
the equation
2
⋅
[
x
]
⋅
{
x
}
=
x
2
−
3
2
⋅
x
−
11
16
2 \cdot [x] \cdot \{x\} = x^2 - \frac32 \cdot x - \frac{11}{16}
2
⋅
[
x
]
⋅
{
x
}
=
x
2
−
2
3
⋅
x
−
16
11
where
[
x
]
[x]
[
x
]
and
{
x
}
\{x\}
{
x
}
represent the integer part and the fractional part of the real number
x
x
x
, respectively.
7
1
Hide problems
program starts with n = 13^{13}
A program works as follows. If the input is given a natural number
n
n
n
(
n
≥
2
n \ge 2
n
≥
2
), then the program consecutively performs the following procedure: it determines the greatest proper divisor of the number
n
n
n
(that is, different from
1
1
1
and
n
n
n
) and subtracts it from the number
n
n
n
, then applies again the same procedure to the obtained result and so on. If the program cannot find any proper divisor of the given number at a step, then it stops and outputs the total number
m
m
m
of procedures performed (this number can be equal to
0
0
0
). The input was given the number
n
=
1
3
13
n = 13^{13}
n
=
1
3
13
. Determine the respective number
m
m
m
at the output.
6
1
Hide problems
min max \sqrt{x(y + 3)} + \sqrt{y(z + 3)} + \sqrt{z(x + 3)} if x + y+ z = 3
The non-negative numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfy the relation
x
+
y
+
z
=
3
x + y+ z = 3
x
+
y
+
z
=
3
. Find the smallest possible numerical value and the largest possible numerical value for the expression
E
(
x
,
y
,
z
)
=
x
(
y
+
3
)
+
y
(
z
+
3
)
+
z
(
x
+
3
)
.
E(x,y, z) = \sqrt{x(y + 3)} + \sqrt{y(z + 3)} + \sqrt{z(x + 3)} .
E
(
x
,
y
,
z
)
=
x
(
y
+
3
)
+
y
(
z
+
3
)
+
z
(
x
+
3
)
.
5
1
Hide problems
\sqrt{n! + 5} is natural
Determine all nonzero natural numbers
n
n
n
, for which the number
n
!
+
5
\sqrt{n! + 5}
n
!
+
5
is a natural number.
4
1
Hide problems
rational m/n = 1+ /2+ 1/3 + ...+ 1/(p-1)
Rational number
m
n
\frac{m}{n}
n
m
admits representation
m
n
=
1
+
1
2
+
1
3
+
.
.
.
+
1
p
−
1
\frac{m}{n} = 1+ \frac12+\frac13 + ...+ \frac{1}{p-1}
n
m
=
1
+
2
1
+
3
1
+
...
+
p
−
1
1
where p
(
p
>
2
)
(p > 2)
(
p
>
2
)
is a prime number. Show that the number
m
m
m
is divisible by
p
p
p
.
2
1
Hide problems
P = product (1 + a_1) is even, independent of arrangement a_i, permutation of n
Let n be the natural number (
n
≥
2
n\ge 2
n
≥
2
). All natural numbers from
1
1
1
up to
n
n
n
,inclusive, are written on the board in some order:
a
1
a_1
a
1
,
a
2
a_2
a
2
,
.
.
.
...
...
,
a
n
a_n
a
n
. Determine all natural numbers
n
n
n
(
n
≥
2
n\ge 2
n
≥
2
), for which the product
P
=
(
1
+
a
1
)
⋅
(
2
+
a
2
)
⋅
.
.
.
⋅
(
n
+
a
n
)
P = (1 + a_1) \cdot (2 + a_2) \cdot ... \cdot (n + a_n)
P
=
(
1
+
a
1
)
⋅
(
2
+
a
2
)
⋅
...
⋅
(
n
+
a
n
)
is an even number, whatever the arrangement of the numbers written on the board.
1
1
Hide problems
(3x+3)/\sqrt{x}} - (x+1)/\sqrt{x^2-x+1} =4
Solve in the set
R
R
R
the equation
3
x
+
3
x
−
x
+
1
x
2
−
x
+
1
=
4
\frac{3x+3}{\sqrt{x}}-\frac{x+1}{\sqrt{x^2-x+1}}=4
x
3
x
+
3
−
x
2
−
x
+
1
x
+
1
=
4
9
1
Hide problems
AB = BP wanted, incenter, tangent at incircle related
The circle inscribed in the triangle
A
B
C
ABC
A
BC
with center
I
I
I
touches the side
B
C
BC
BC
at the point
D
D
D
. The line
D
I
DI
D
I
intersects the side
A
C
AC
A
C
at the point
M
M
M
. The tangent from
M
M
M
to the inscribed circle, different from
A
C
AC
A
C
, intersects the side
A
B
AB
A
B
at the point
N
N
N
. The line
N
I
NI
N
I
intersects the side
B
C
BC
BC
at the point
P
P
P
. Prove that
A
B
=
B
P
AB = BP
A
B
=
BP
.
8
1
Hide problems
<BAC + <MIN = 180^o, incenter, AC = NC, AB = MB
Let
A
B
C
ABC
A
BC
be the triangle and
I
I
I
the center of the circle inscribed in this triangle. The point
M
M
M
, located on the tangent taken to the point
B
B
B
to the circumscribed circle of the triangle
A
B
C
ABC
A
BC
, satisfies the relation
A
B
=
M
B
AB = MB
A
B
=
MB
. Point
N
N
N
, located on the tangent taken to point
C
C
C
to the same circle, satisfies the relation
A
C
=
N
C
AC = NC
A
C
=
NC
. Points
M
,
A
M, A
M
,
A
and
N
N
N
lie on the same side of the line
B
C
BC
BC
. Prove that
∠
B
A
C
+
∠
M
I
N
=
18
0
o
.
\angle BAC + \angle MIN = 180^o.
∠
B
A
C
+
∠
M
I
N
=
18
0
o
.
3
1
Hide problems
junior collinearity, intersecting circles related
Circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
intersect at points
A
A
A
and
B
B
B
. A straight line is drawn through point
B
B
B
, which again intersects circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
at points
C
C
C
and
D
D
D
, respectively. Point
E
E
E
, located on circle
ω
1
\omega_1
ω
1
, satisfies the relation
C
E
=
C
B
CE = CB
CE
=
CB
, and point
F
F
F
, located on circle
ω
2
\omega_2
ω
2
, satisfies the relation
D
B
=
D
F
DB = DF
D
B
=
D
F
. The line
B
F
BF
BF
intersects again the circle
ω
1
\omega_1
ω
1
at the point
P
P
P
, and the line
B
E
BE
BE
intersects again the circle
ω
2
\omega_2
ω
2
at the point
Q
Q
Q
. Prove that the points
A
,
P
A, P
A
,
P
, and
Q
Q
Q
are collinear.