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Polish Junior Math Olympiad
2022 Polish Junior Math Olympiad
2022 Polish Junior Math Olympiad Second Round
2022 Polish Junior Math Olympiad Second Round
Part of
2022 Polish Junior Math Olympiad
Subcontests
(5)
1.
1
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2022 Polish Junior Math Olympiad Second Round
The line segments
A
B
AB
A
B
and
C
D
CD
C
D
are perpendicular and intersect at point
X
X
X
. Additionally, the following equalities hold:
A
C
=
B
D
AC=BD
A
C
=
B
D
,
A
D
=
B
X
AD=BX
A
D
=
BX
, and
D
X
=
1
DX=1
D
X
=
1
. Determine the length of segment
C
X
CX
CX
.
2.
1
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2022 Polish Junior Math Olympiad Second Round P2
Let
n
≥
1
n\geq 1
n
≥
1
be an integer and let
a
a
a
and
b
b
b
be its positive divisors satisfying
a
+
b
+
a
b
=
n
a+b+ab=n
a
+
b
+
ab
=
n
. Prove that
a
=
b
a=b
a
=
b
.
3.
1
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2022 Polish Junior Math Olympiad Second Round P3
Let
n
n
n
be a positive integer. Each of the numbers
1
,
2
,
3
,
…
,
100
1,2,3,\ldots,100
1
,
2
,
3
,
…
,
100
is painted with one of
n
n
n
colors in such a way that two distinct numbers with a sum divisible by
4
4
4
are painted with different colors. Determine the smallest value of
n
n
n
for which such a situation is possible.
4.
1
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2022 Polish Junior Math Olympiad Second Round P4
In the convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
, the following equalities hold:
∠
C
D
E
=
9
0
∘
\angle CDE=90^\circ
∠
C
D
E
=
9
0
∘
,
A
C
=
A
D
AC=AD
A
C
=
A
D
, and
B
D
=
B
E
BD=BE
B
D
=
BE
. Prove that triangle
A
B
D
ABD
A
B
D
and quadrilateral
A
B
C
E
ABCE
A
BCE
have the same area.
5.
1
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2022 Polish Junior Math Olympiad Second Round P5
Let
n
≥
3
n\geq 3
n
≥
3
be an odd integer. On a line,
n
n
n
points are marked in such a way that the distance between any two of them is an integer. It turns out that each marked point has an even sum of distances to the remaining
n
−
1
n-1
n
−
1
marked points. Prove that the distance between any two marked points is even.