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Polish Junior Math Olympiad
2022 Polish Junior Math Olympiad
2022 Polish Junior Math Olympiad First Round
2022 Polish Junior Math Olympiad First Round
Part of
2022 Polish Junior Math Olympiad
Subcontests
(7)
1.
1
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2022 Polish Junior Math Olympiad First Round P1
There are
17
17
17
students in Marek's class, and all of them took a test. Marek's score was
17
17
17
points higher than the arithmetic mean of the scores of the other students. By how many points is Marek's score higher than the arithmetic mean of the scores of the entire class? Justify your answer.
2.
1
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2022 Polish Junior Math Olympiad First Round P2
In the rectangle
A
B
C
D
ABCD
A
BC
D
, the ratio of the lengths of sides
B
C
BC
BC
and
A
B
AB
A
B
is equal to
2
\sqrt{2}
2
. Point
X
X
X
is marked inside this rectangle so that
A
B
=
B
X
=
X
D
AB=BX=XD
A
B
=
BX
=
X
D
. Determine the measure of angle
B
X
D
BXD
BX
D
.
3.
1
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2022 Polish Junior Math Olympiad First Round P3
Let
n
≥
1
n\geq 1
n
≥
1
be an integer. Show that there exists an integer between
2
n
\sqrt{2n}
2
n
and
5
n
\sqrt{5n}
5
n
, exclusive.
4.
1
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2022 Polish Junior Math Olympiad First Round P4
In each square of the table below, we must write a different integer from
1
1
1
to
17
17
17
, such that the sum of the numbers in each of the eight columns is the same, and the sum of the numbers in the top row is twice the sum of the numbers in the bottom row. Which number from
1
1
1
to
17
17
17
can be omitted? https://wiki-images.artofproblemsolving.com//2/2b/Zrzut_ekranu_2023-05-22_o_10.28.33.png
5.
1
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2022 Polish Junior Math Olympiad First Round P5
Points
K
K
K
,
L
L
L
,
M
M
M
lie on the sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
of equilateral triangle
A
B
C
ABC
A
BC
respectively, and satisfy the conditions
K
M
=
L
M
KM=LM
K
M
=
L
M
,
∠
K
M
L
=
9
0
∘
\angle KML=90^\circ
∠
K
M
L
=
9
0
∘
, and
A
M
=
B
K
AM=BK
A
M
=
B
K
. Prove that
∠
C
K
L
=
9
0
∘
\angle CKL=90^\circ
∠
C
K
L
=
9
0
∘
.
6.
1
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2022 Polish Junior Math Olympiad First Round P6
In each square of a
10
×
10
10\times 10
10
×
10
board, there is an arrow pointing upwards, downwards, left, or right. Prove that it is possible to remove
50
50
50
arrows from the board, such that no two remaining arrows point at each other.
7.
1
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2022 Polish Junior Math Olympiad First Round P7
None of the
n
n
n
(not necessarily distinct) digits selected are equal to
0
0
0
or
7
7
7
. It turns out that every
n
n
n
-digit number formed using these digits is divisible by
7
7
7
. Prove that
n
n
n
is divisible by
6
6
6
.