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Problems
Contests
National and Regional Contests
Poland Contests
Polish Junior Math Olympiad
2018 Polish Junior Math Olympiad
2018 Polish Junior MO First Round
2018 Polish Junior MO First Round
Part of
2018 Polish Junior Math Olympiad
Subcontests
(7)
7
1
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Plane cuts cuboid, calculate volume of blocks
Square
A
B
C
D
ABCD
A
BC
D
with sides of length
4
4
4
is a base of a cuboid
A
B
C
D
A
′
B
′
C
′
D
′
ABCDA'B'C'D'
A
BC
D
A
′
B
′
C
′
D
′
. Side edges
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
,
C
C
′
CC'
C
C
′
,
D
D
′
DD'
D
D
′
of this cuboid have length
7
7
7
. Points
K
,
L
,
M
K, L, M
K
,
L
,
M
lie respectively on line segments
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
,
C
C
′
CC'
C
C
′
, and
A
K
=
3
AK = 3
A
K
=
3
,
B
L
=
2
BL = 2
B
L
=
2
,
C
M
=
5
CM = 5
CM
=
5
. Plane passing through points
K
,
L
,
M
K, L, M
K
,
L
,
M
cuts cuboid on two blocks. Calculate volumes of these blocks.
5
1
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Coloring integers by three colors
Each integer should be colored by one of three colors, including red. Each number which can be represent as a sum of two numbers of different colors should be red. Each color should be used. Is this coloring possible?
6
1
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Integers, equation
Positive integers
k
,
m
,
n
k, m, n
k
,
m
,
n
satisfy the equation
m
2
+
n
=
k
2
+
k
m^2 + n = k^2 + k
m
2
+
n
=
k
2
+
k
. Show that
m
≤
n
m \le n
m
≤
n
.
4
1
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Equality of angles in trapezoid
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid with bases
A
B
AB
A
B
and
C
D
CD
C
D
. Bisectors of
A
D
AD
A
D
and
B
C
BC
BC
intersect line segments
B
C
BC
BC
and
A
D
AD
A
D
respectively in points
P
P
P
and
Q
Q
Q
. Show that
∠
A
P
D
=
∠
B
Q
C
\angle APD = \angle BQC
∠
A
P
D
=
∠
BQC
.
3
1
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Show that product is divisible by 48
Prime numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
are bigger that
3
3
3
. Show that
(
a
−
b
)
(
b
−
c
)
(
c
−
a
)
(a - b)(b - c)(c - a)
(
a
−
b
)
(
b
−
c
)
(
c
−
a
)
is divisible by
48
48
48
.
2
1
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Parallelogram, perpendicular lines
Inside parallelogram
A
B
C
D
ABCD
A
BC
D
is point
P
P
P
, such that
P
C
=
B
C
PC = BC
PC
=
BC
. Show that line
B
P
BP
BP
is perpendicular to line which connects middles of sides of line segments
A
P
AP
A
P
and
C
D
CD
C
D
.
1
1
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Show pythagorean equality
Numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
are such that
3
a
+
4
b
=
3
c
3a + 4b = 3c
3
a
+
4
b
=
3
c
and
4
a
−
3
b
=
4
c
4a - 3b = 4c
4
a
−
3
b
=
4
c
. Show that
a
2
+
b
2
=
c
2
a^2 + b^2 = c^2
a
2
+
b
2
=
c
2
.