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Polish Junior Math Olympiad
2023 Polish Junior Math Olympiad
2023 Polish Junior Math Olympiad First Round
2023 Polish Junior Math Olympiad First Round
Part of
2023 Polish Junior Math Olympiad
Subcontests
(7)
6.
1
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2023 Polish Junior Math Olympiad Round 1 P6
We call the figure shown in the picture consisting of five unit squares a \emph{plus}, and each rectangle consisting of two such squares a \emph{minus}. Does there exist an odd integer
n
n
n
with the property that a square with side length
n
n
n
can be dissected into pluses and minuses? Justify your answer. https://wiki-images.artofproblemsolving.com//6/6a/18-1-6.png
7.
1
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2023 Polish Junior Math Olympiad Round 1 P7
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a regular hexagon with side length
2
2
2
. Point
M
M
M
is the midpoint of diagonal
A
E
AE
A
E
. The pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
is folded along segments
B
D
BD
B
D
,
B
M
BM
BM
, and
D
M
DM
D
M
in such a way that points
A
A
A
,
C
C
C
, and
E
E
E
coincide. As a result of this operation, a tetrahedron is obtained. Determine its volume.
5.
1
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2023 Polish Junior Math Olympiad Round 1 P5
Positive numbers
a
a
a
,
b
b
b
,
c
c
c
satisfy the inequalities a + b \geq ab, b + c \geq bc, \text{and} c+ a \geq ca. Prove that
a
+
b
+
c
≥
3
4
a
b
c
\displaystyle a + b + c \geq \frac34abc
a
+
b
+
c
≥
4
3
ab
c
.
4.
1
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2023 Polish Junior Math Olympiad Round 1 P4
Each of the natural numbers from
1
1
1
to
n
n
n
is colored either red or blue, with each color being used at least once. It turns out that: – every red number is a sum of two distinct blue numbers; and – every blue number is a difference between two red numbers. Determine the smallest possible value of
n
n
n
for which such a coloring exists.
3.
1
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2023 Polish Junior Math Olympiad Round 1 P3
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle. Point
E
E
E
lies on side
A
B
AB
A
B
, and point
F
F
F
lies on segment
C
E
CE
CE
. Prove that if triangles
A
D
E
ADE
A
D
E
and
C
D
F
CDF
C
D
F
have equal areas, then triangles
B
C
E
BCE
BCE
and
D
E
F
DEF
D
EF
also have equal areas.
2.
1
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2023 Polish Junior Math Olympiad Round 1 P2
Kamil wrote on a board an expression consisting of alternating addition and subtraction of natural numbers from
1
1
1
to
100
100
100
:
1
−
2
+
3
−
4
+
5
−
6
+
…
−
98
+
99
−
100.
1-2+3-4+5-6+\ldots-98+99-100.
1
−
2
+
3
−
4
+
5
−
6
+
…
−
98
+
99
−
100.
Then, Kamil erased one of the plus or minus signs and replaced it with an equals sign, obtaining a true equality. Which number preceded the erased sign? Find all possibilities and justify your answer.
1.
1
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2023 Polish Junior Math Olympiad Round 1 P1
Given is a rectangle with perimeter
x
x
x
cm and side lengths in a
1
:
2
1:2
1
:
2
ratio. Suppose that the area of the rectangle is also
x
x
x
cm
2
\text{cm}^2
cm
2
. Determine all possible values of
x
x
x
.