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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 Second Round
2018 Polish Junior MO Second Round
Part of
2018 Polish Junior Math Olympiad
Subcontests
(5)
5
1
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Coloring of integers
Each integer has been colored in one of three colors. Prove that exist two different numbers of the same color, whose difference is a perfect square.
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
. Points
P
P
P
and
Q
Q
Q
lie on diagonals
A
C
AC
A
C
and
B
D
BD
B
D
, respectively and
∠
A
P
D
=
∠
B
Q
C
\angle APD = \angle BQC
∠
A
P
D
=
∠
BQC
. Prove that
∠
A
Q
D
=
∠
B
P
C
\angle AQD = \angle BPC
∠
A
Q
D
=
∠
BPC
.
3
1
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System of equations
Determine all trios of integers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
which are solution of system of equations
{
x
−
y
z
=
1
x
z
+
y
=
2
\begin{cases} x - yz = 1 \\ xz + y = 2 \end{cases}
{
x
−
yz
=
1
x
z
+
y
=
2
2
1
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Acute triangle, equality of areas
Let
A
B
C
ABC
A
BC
be an acute traingle with
A
C
≠
B
C
AC \neq BC
A
C
=
BC
. Point
K
K
K
is a foot of altitude through vertex
C
C
C
. Point
O
O
O
is a circumcenter of
A
B
C
ABC
A
BC
. Prove that areas of quadrilaterals
A
K
O
C
AKOC
A
K
OC
and
B
K
O
C
BKOC
B
K
OC
are equal.
1
1
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Do positive reals exist?
Do positive reals
a
,
b
,
c
,
x
a, b, c, x
a
,
b
,
c
,
x
such that
a
2
+
b
2
=
c
2
a^2+ b^2 = c^2
a
2
+
b
2
=
c
2
and
(
a
+
x
)
2
+
(
b
+
x
)
2
=
(
c
+
x
)
2
(a + x)^2+ (b +x)^2 = (c + x)^2
(
a
+
x
)
2
+
(
b
+
x
)
2
=
(
c
+
x
)
2
exist?