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National and Regional Contests
Poland Contests
Polish Junior Math Olympiad
2017 Polish Junior Math Olympiad
2017 Polish Junior Math Olympiad First Round
2017 Polish Junior Math Olympiad First Round
Part of
2017 Polish Junior Math Olympiad
Subcontests
(7)
1.
1
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2017 Polish Junior Math Olympiad First Round P1
Rational numbers
a
a
a
,
b
b
b
,
c
c
c
satisfy the equation
(
a
+
b
+
c
)
(
a
+
b
−
c
)
=
c
2
.
(a+b+c)(a+b-c)=c^2\,.
(
a
+
b
+
c
)
(
a
+
b
−
c
)
=
c
2
.
Show that
a
+
b
=
c
=
0
a+b=c=0
a
+
b
=
c
=
0
.
2.
1
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2017 Polish Junior Math Olympiad First Round P2
Consider an acute triangle
A
B
C
ABC
A
BC
with
∠
A
C
B
=
4
5
∘
.
\angle ACB=45^\circ\,.
∠
A
CB
=
4
5
∘
.
Let
B
C
E
D
BCED
BCE
D
and
A
C
F
G
ACFG
A
CFG
be squares lying outside triangle
A
B
C
ABC
A
BC
. Prove that the midpoint of segment
D
G
DG
D
G
coincides with the circumcenter of triangle
A
B
C
ABC
A
BC
.
3.
1
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2017 Polish Junior Math Olympiad First Round P3
In each square of an
11
×
11
11\times 11
11
×
11
board, we are to write one of the numbers
−
1
-1
−
1
,
0
0
0
, or
1
1
1
in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.
4.
1
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2017 Polish Junior Math Olympiad First Round P4
Quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in a circle with
∠
A
B
C
=
6
0
∘
\angle ABC=60^\circ
∠
A
BC
=
6
0
∘
and
B
C
=
C
D
BC=CD
BC
=
C
D
. Prove that
A
B
=
A
D
+
D
C
AB=AD+DC
A
B
=
A
D
+
D
C
.
5.
1
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2017 Polish Junior Math Olympiad First Round P5
Let
a
a
a
and
b
b
b
be the positive integers. Show that at least one of the numbers
a
a
a
,
b
b
b
,
a
+
b
a+b
a
+
b
can be expressed as the difference of the squares of two integers.
6.
1
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2017 Polish Junior Math Olympiad First Round P6
The base of the pyramid
A
B
C
D
ABCD
A
BC
D
is an equilateral triangle
A
B
C
ABC
A
BC
with side length
1
1
1
. Additionally,
∠
A
D
B
=
∠
B
D
C
=
∠
C
D
A
=
9
0
∘
.
\angle ADB=\angle BDC=\angle CDA=90^\circ\,.
∠
A
D
B
=
∠
B
D
C
=
∠
C
D
A
=
9
0
∘
.
Calculate the volume of pyramid
A
B
C
D
ABCD
A
BC
D
.
7.
1
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2017 Polish Junior Math Olympiad First Round P7
Let
a
a
a
and
b
b
b
be positive integers such that the prime number
a
+
b
+
1
a+b+1
a
+
b
+
1
divides the integer
4
a
b
−
1
4ab-1
4
ab
−
1
. Prove that
a
=
b
a=b
a
=
b
.