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Contests
National and Regional Contests
Moldova Contests
JBMO TST - Moldova
2024 Junior Balkan Team Selection Tests - Moldova
2024 Junior Balkan Team Selection Tests - Moldova
Part of
JBMO TST - Moldova
Subcontests
(8)
11
1
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2024x2023 rectangle is filled with 3 types of pieces
A rectangle of dimensions
2024
×
2023
2024 \times 2023
2024
×
2023
is filled with pieces of the following types: [asy]size(200);// Figure (A) draw((0,0)--(4,0)--(4,1)--(0,1)--cycle); draw((1,0)--(1,1)); draw((2,0)--(2,1)); draw((3,0)--(3,1));// Figure (B) draw((6,0)--(8,0)--(8,2)--(6,2)--cycle); draw((7,0)--(7,2)); draw((6,1)--(8,1));// Figure (C) draw((10,0)--(12,0)--(12,1)--(11,1)--(11,2)--(9,2)--(9,1)--(10,1)--cycle); draw((10,0)--(10,1)); draw((11,0)--(11,1)); draw((10,1)--(11,1)); draw((9,1)--(9,2)); draw((10,1)--(10,2)); draw((11,0)--(12,0)); draw((10,1)--(12,1));// Labeling label("(A)", (2, -0.5)); label("(B)", (7, -0.5)); label("(C)", (10.5, -0.5));[/asy] Each piece can be turned arround, and each square has side length
1
1
1
. Is it possible to use exactly 2023 pieces of type
(
A
)
(A)
(
A
)
?
9
1
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equal parallelograms' areas
Consider the parallelograms
A
B
C
D
ABCD
A
BC
D
and
A
X
Y
Z
AXYZ
A
X
Y
Z
, such that
X
∈
X \in
X
∈
[
B
C
BC
BC
] and
D
∈
D \in
D
∈
[
Y
Z
YZ
Y
Z
]. Prove that the areas of the parallelograms are equal.
7
1
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inequality hidden behind an equality
Find all the real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
which satisfy the following conditions:
{
3
(
x
2
+
y
2
+
z
2
)
=
1
x
2
y
2
+
y
2
z
2
+
z
2
x
2
=
x
y
z
(
x
+
y
+
z
)
3
\begin{cases} 3(x^2+y^2+z^2)=1\\ x^2y^2+y^2z^2+z^2x^2=xyz(x+y+z)^3\\ \end{cases}
{
3
(
x
2
+
y
2
+
z
2
)
=
1
x
2
y
2
+
y
2
z
2
+
z
2
x
2
=
x
yz
(
x
+
y
+
z
)
3
6
1
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Circumcenter of XYZ lies on AC
In the isosceles triangle
A
B
C
ABC
A
BC
, with
A
B
=
B
C
AB=BC
A
B
=
BC
, points
X
X
X
and
Y
Y
Y
are the midpoints of the sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. Point
Z
Z
Z
is the foot of the perpendicular from
B
B
B
to
C
X
CX
CX
. Prove that the circumcenter of the triangle
X
Y
Z
XYZ
X
Y
Z
is of the line
A
C
AC
A
C
.
5
1
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no numbers of the form 80...01 are squares
Prove that a number of the form
80
…
01
80\dots01
80
…
01
(there is at least 1 zero) can't be a perfect square.
3
1
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999 points in the plane
Let
M
M
M
be a set of 999 points in the plane with the property: For any 3 distinct points in
M
M
M
we can choose two of them, such that the distance between them is less than
1
1
1
. a)Prove that there exists a disc of radius not greater than 1 that covers at least 500 points in
M
M
M
. b)Is it true that there always exists a disc of radius not greater than 1 that covers at least 501 points in
M
M
M
?
2
1
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2024 8's
Prove that the number
88
…
8
⏟
2024
times
\underbrace{88\dots8}_\text{2024\; \textrm{times}}
2024
times
88
…
8
is divisible by 2024.
1
1
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a+b+c=xyz=1 inequality in 6 variables
Let
a
,
b
,
c
,
x
,
y
,
z
a,b,c,x,y,z
a
,
b
,
c
,
x
,
y
,
z
be positive real numbers, such that
a
+
b
+
c
=
x
y
z
=
1
a+b+c=xyz=1
a
+
b
+
c
=
x
yz
=
1
Prove that:
x
2
3
a
+
2
+
y
2
3
b
+
2
+
z
2
3
c
+
2
≥
1
\frac{x^2}{3a+2}+\frac{y^2}{3b+2}+\frac{z^2}{3c+2} \ge 1
3
a
+
2
x
2
+
3
b
+
2
y
2
+
3
c
+
2
z
2
≥
1
When does equality hold?