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Polish Junior Math Olympiad
2017 Polish Junior Math Olympiad
2017 Polish Junior Math Olympiad Finals
2017 Polish Junior Math Olympiad Finals
Part of
2017 Polish Junior Math Olympiad
Subcontests
(5)
1.
1
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2017 Polish Junior Math Olympiad Finals P1
Let
a
a
a
,
b
b
b
, and
c
c
c
be positive integers for which the number
a
2
+
b
b
2
+
c
\frac{a\sqrt2+b}{b\sqrt2+c}
b
2
+
c
a
2
+
b
is rational. Show that the number
a
b
+
b
c
+
c
a
ab+bc+ca
ab
+
b
c
+
c
a
is divisible by
a
+
b
+
c
a+b+c
a
+
b
+
c
.
2.
1
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2017 Polish Junior Math Olympiad Finals P2
Point
D
D
D
lies on the side
A
B
AB
A
B
of triangle
A
B
C
ABC
A
BC
, and point
E
E
E
lies on the segment
C
D
CD
C
D
. Prove that if the sum of the areas of triangles
A
C
E
ACE
A
CE
and
B
D
E
BDE
B
D
E
is equal to half the area of triangle
A
B
C
ABC
A
BC
, then either point
D
D
D
is the midpoint of
A
B
AB
A
B
or point
E
E
E
is the midpoint of
C
D
CD
C
D
.
3.
1
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2017 Polish Junior Math Olympiad Finals P3
Positive integers
a
a
a
and
b
b
b
are given such that each of the numbers
a
b
ab
ab
and
(
a
+
1
)
(
b
+
1
)
(a+1)(b+1)
(
a
+
1
)
(
b
+
1
)
is a perfect square. Prove that there exists an integer
n
>
1
n>1
n
>
1
such that the number
(
a
+
n
)
(
b
+
n
)
(a+n)(b+n)
(
a
+
n
)
(
b
+
n
)
is a perfect square.
4.
1
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2017 Polish Junior Math Olympiad Finals P4
In the convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
, the angles at the vertices
B
B
B
,
C
C
C
,
E
E
E
, and
F
F
F
are equal. Moreover, the equality
A
B
+
D
E
=
A
F
+
C
D
AB+DE=AF+CD
A
B
+
D
E
=
A
F
+
C
D
holds. Prove that the line
A
D
AD
A
D
and the bisectors of the segments
B
C
BC
BC
and
E
F
EF
EF
have a common point.
5.
1
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2017 Polish Junior Math Olympiad Finals P5
There are
n
n
n
matches lying on a table, forming
n
n
n
one-match piles. Adam wants to combine them into a single pile of
n
n
n
matches. He will do this using
n
−
1
n-1
n
−
1
operations, each of which consists of combining two piles into one. Adam has made a deal with Bartek that every time he combines a pile of
a
a
a
matches with a pile of
b
b
b
matches, he will receive
a
⋅
b
a\cdot b
a
⋅
b
candies from Bartek. What is the greatest number of candies that Adam can receive after performing
n
−
1
n-1
n
−
1
operations? Justify your answer.