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Problems
Contests
National and Regional Contests
Poland Contests
Polish Junior Math Olympiad
2019 Polish Junior Math Olympiad
2019 Polish Junior MO Finals
2019 Polish Junior MO Finals
Part of
2019 Polish Junior Math Olympiad
Subcontests
(5)
5.
1
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Board and numbers
In the every cell of the board
5
×
5
5\times5
5
×
5
there is one of the numbers:
−
1
-1
−
1
,
0
0
0
,
1
1
1
. It is true that in every
2
×
2
2 \times 2
2
×
2
square there are three numbers summing up to
0
0
0
. Determine the maximal sum of all numbers in a board.
4.
1
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Beautiful geometric inequality
The point
D
D
D
lies on the side
A
B
AB
A
B
of the triangle
A
B
C
ABC
A
BC
. Assume that there exists such a point
E
E
E
on the side
C
D
CD
C
D
, that \sphericalangle EAD = \sphericalangle AED \text{and} \sphericalangle ECB = \sphericalangle CEB. Show that
A
C
+
B
C
>
A
B
+
C
E
AC + BC > AB + CE
A
C
+
BC
>
A
B
+
CE
.
3.
1
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Algebra, fractions and numbers
Let
x
x
x
,
y
y
y
,
z
z
z
be non-zero real numbers, such that
x
+
y
+
z
=
0
x + y + z = 0
x
+
y
+
z
=
0
and the numbers \frac{x}{y} + \frac{y}{z} + \frac{z}{x} \text{and} \frac{x}{z} + \frac{z}{y} + \frac{y}{x} + 1 are equal. Determine their common value.
2.
1
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Trapezium and isosceles triangles
Let
A
B
C
D
ABCD
A
BC
D
be the isosceles trapezium with bases
A
B
AB
A
B
and
C
D
CD
C
D
, such that
A
C
=
B
C
AC = BC
A
C
=
BC
. The point
M
M
M
is the midpoint of side
A
D
AD
A
D
. Prove that
∢
A
C
M
=
∢
C
B
D
\sphericalangle ACM = \sphericalangle CBD
∢
A
CM
=
∢
CB
D
.
1.
1
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Two integers
Let
a
a
a
,
b
b
b
be the positive integers greater than
1
1
1
. Prove that if
a
b
,
a
−
1
b
−
1
\frac{a}{b},\; \frac{a-1}{b-1}
b
a
,
b
−
1
a
−
1
differ by 1, then both are integers.