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Polish Junior Math Olympiad
2022 Polish Junior Math Olympiad
2022 Polish Junior Math Olympiad Finals
2022 Polish Junior Math Olympiad Finals
Part of
2022 Polish Junior Math Olympiad
Subcontests
(5)
1.
1
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2022 Polish Junior Math Olympiad Finals P1
Given is a square
A
B
C
D
ABCD
A
BC
D
with side length
1
1
1
. Points
K
K
K
,
L
L
L
,
M
M
M
, and
N
N
N
, distinct from the vertices of the square, lie on segments
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
, and
D
A
DA
D
A
, respectively. Prove that the perimeter of at least one of the triangles
A
N
K
ANK
A
N
K
,
B
K
L
BKL
B
K
L
,
C
L
M
CLM
C
L
M
,
D
M
N
DMN
D
MN
is less than
2
2
2
.
2.
1
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2022 Polish Junior Math Olympiad Finals P2
Find all positive integers
n
n
n
for which both numbers 1\;\;\!\!\!\!\underbrace{77\ldots 7}_{\text{$n$ sevens}}\!\!\!\! \text{and} 3\;\; \!\!\!\!\underbrace{77\ldots 7}_{\text{$n$ sevens}} are prime.
3.
1
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2022 Polish Junior Math Olympiad Finals P3
Given a parallelogram
A
B
C
D
ABCD
A
BC
D
in which
∠
A
B
D
=
9
0
∘
\angle ABD=90^\circ
∠
A
B
D
=
9
0
∘
and
∠
C
B
D
=
4
5
∘
\angle CBD=45^\circ
∠
CB
D
=
4
5
∘
. Point
E
E
E
lies on segment
A
D
AD
A
D
such that
B
C
=
C
E
BC=CE
BC
=
CE
. Determine the measure of angle
B
C
E
BCE
BCE
.
4.
1
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2022 Polish Junior Math Olympiad Finals P4
Find all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of nonzero integers for which
(
1
−
a
)
(
1
−
b
)
(
1
−
c
)
=
(
1
+
a
)
(
1
+
b
)
(
1
+
c
)
.
(1-a)(1-b)(1-c)=(1+a)(1+b)(1+c).
(
1
−
a
)
(
1
−
b
)
(
1
−
c
)
=
(
1
+
a
)
(
1
+
b
)
(
1
+
c
)
.
5.
1
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2022 Polish Junior Math Olympiad Finals P5
In the table shown in the figure, Zosia replaced eight numbers with their negatives. It turned out that each row and each column contained exactly two negative numbers. Prove that after this change, the sum of all sixteen numbers in the table is equal to
0
0
0
. https://wiki-images.artofproblemsolving.com//2/2e/17-3-5.png