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Contests
National and Regional Contests
Poland Contests
Polish Junior Math Olympiad
2023 Polish Junior Math Olympiad
2023 Polish Junior MO Second Round
2023 Polish Junior MO Second Round
Part of
2023 Polish Junior Math Olympiad
Subcontests
(5)
5.
1
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Combinatorics with table
In each cell of a
4
×
4
4\times 4
4
×
4
table, one of the numbers
1
1
1
or
2
2
2
is written. For each row, calculate the sum of the numbers written in it, and for each column, calculate the product of the numbers written in it. Show that some two of the eight results obtained are equal.
4.
1
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Geometry with parallelogram
Consider a parallelogram
A
B
C
D
ABCD
A
BC
D
where
A
B
>
A
D
AB>AD
A
B
>
A
D
. Let
X
X
X
and
Y
Y
Y
, distinct from
B
B
B
, be points on the ray
B
D
→
BD^\rightarrow
B
D
→
such that
C
X
=
C
B
CX=CB
CX
=
CB
and
A
Y
=
A
B
AY=AB
A
Y
=
A
B
. Prove that
D
X
=
D
Y
DX=DY
D
X
=
D
Y
. Note: The notation
B
D
→
BD^\rightarrow
B
D
→
denotes the ray originating from point
B
B
B
passing through point
D
D
D
.
3.
1
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Number theory with digits
A natural number
n
n
n
is at least two digits long. If we write a certain digit between the tens digit and the units digit of this number, we obtain six times the number
n
n
n
. Find all numbers
n
n
n
with this property.
2.
1
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Two integers on the board
Initially, the numbers
2
2
2
and
5
5
5
are written on the board. A \emph{move} consists of replacing one of the two numbers on the board with their sum. Is it possible to obtain (in a finite numer of moves) a situation in which the two integers written on the board are consecutive? Justify your answer.
1.
1
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Angles and sides equal.
On the sides
A
B
AB
A
B
and
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
, there are points
D
D
D
and
E
E
E
, respectively, such that \angle ADC=\angle BDE \text{and} \angle BCD=\angle AED. Prove that
A
E
=
B
E
AE=BE
A
E
=
BE
.