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Problems
Contests
National and Regional Contests
Poland Contests
Polish Junior Math Olympiad
2023 Polish Junior Math Olympiad
2023 Polish Junior Math Olympiad Finals
2023 Polish Junior Math Olympiad Finals
Part of
2023 Polish Junior Math Olympiad
Subcontests
(5)
2.
1
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Divisibility implies non-power
There are integers
a
a
a
and
b
b
b
, such that
a
>
b
>
1
a>b>1
a
>
b
>
1
and
b
b
b
is the largest divisor of
a
a
a
different from
a
a
a
. Prove that the number
a
+
b
a+b
a
+
b
is not a power of
2
2
2
with integer exponent.
5.
1
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Digits and divisibility
Find all pairs of positive integers
m
m
m
,
n
n
n
such that the
(
m
+
n
)
(m+n)
(
m
+
n
)
-digit number
33
…
3
⏟
m
66
…
6
⏟
n
\underbrace{33\ldots3}_{m}\underbrace{66\ldots 6}_{n}
m
33
…
3
n
66
…
6
is a perfect square.
4.
1
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Combinatorics with arrows
Let
n
≥
1
n\geq 1
n
≥
1
be odd integer. There are
n
n
n
arrows are arranged from left to right, such that each arrow points either to the left or to the right. Prove that there exists an arrow that is pointed to by exactly as many arrows as it is pointing to. Note: For example, for
n
=
5
n=5
n
=
5
and the arrangement
→
→
←
←
→
\rightarrow\rightarrow\leftarrow\leftarrow\rightarrow
→→←←→
, the successive arrows (from the left) point respectively to
4
4
4
,
3
3
3
,
2
2
2
,
3
3
3
,
0
0
0
arrows.
3.
1
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Around equilateral triangles
Triangle
A
B
C
ABC
A
BC
is given, where
A
C
<
B
C
AC<BC
A
C
<
BC
and
∠
A
C
B
=
6
0
∘
.
\angle ACB=60^\circ\!\!.
∠
A
CB
=
6
0
∘
.
Point
D
D
D
, distinct from
A
A
A
, lies on the segment
A
C
AC
A
C
such that
A
B
=
B
D
AB=BD
A
B
=
B
D
, and point
E
E
E
, distinct from
B
B
B
, lies on the line
B
C
BC
BC
such that
A
B
=
A
E
AB=AE
A
B
=
A
E
. Prove that
∠
D
E
C
=
3
0
∘
\angle DEC=30^\circ
∠
D
EC
=
3
0
∘
.
1.
1
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Cyclic System of Equations
Determine whether there exist real numbers
x
x
x
,
y
y
y
,
z
z
z
, such that x+\frac{1}{y}=z, y+\frac{1}{z}=x, z+\frac{1}{x}=y.