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Problems
Contests
National and Regional Contests
Poland Contests
Polish Junior Math Olympiad
2021 Polish Junior Math Olympiad
2021 Polish Junior MO Second Round
2021 Polish Junior MO Second Round
Part of
2021 Polish Junior Math Olympiad
Subcontests
(5)
5
1
Hide problems
among invited guests, 1 know's all other 10 guests (2021 Polish JMO R2 p5)
Tomek invited to a remote birthday part
11
11
11
of his friends who will join the meeting one by one. Tomek chose the guests in such a way that, regardless of the order in which they will join, always the newcomer knew at least half of the people already present, including Tomek. Prove that among of invited guests, there is one who knows all of Tom's other
10
10
10
friends. Caution: We assume that if person A knows person
B
B
B
, then
B
B
B
also knows
A
A
A
.[hide=original wording]Tomek zaprosił na zdalne przyjęcie urodzinowe 11 swoich znajomych, którzy kolejno będą dołączać do spotkania. Tomek dobrał gości w taki sposób, aby niezależnie od kolejności w jakiej będą dołączać, zawsze nowo przybyła osoba znała co najmniej połowę już obecnych osób, wliczając Tomka. Wykaż, że wśród zaproszonych gości istnieje taki, który zna wszystkich pozostałych 10 znajomych Tomka.Uwaga: Przyjmujemy, że jeśli osoba A zna osobę B, to również B zna A.
4
1
Hide problems
angle bisector of <BAD _|_ KL, AB+BK = AD+DL, #ABCD (2021 Polish JMO R2 p4)
Points
K
K
K
and
L
L
L
are on the sides
B
C
BC
BC
and
C
D
CD
C
D
, respectively of the parallelogram
A
B
C
D
ABCD
A
BC
D
, such that
A
B
+
B
K
=
A
D
+
D
L
AB + BK = AD + DL
A
B
+
B
K
=
A
D
+
D
L
. Prove that the bisector of angle
B
A
D
BAD
B
A
D
is perpendicular to the line
K
L
KL
K
L
.
3
1
Hide problems
pos. integers a=b if (5a + 3b) is divisible by (a + b) (2021 Polish JMO R2 p3)
Given are positive integers
a
,
b
a, b
a
,
b
for which
5
a
+
3
b
5a + 3b
5
a
+
3
b
is divisible by
a
+
b
a + b
a
+
b
. Prove that
a
=
b
a = b
a
=
b
.
2
1
Hide problems
<DEF = 90^o wanted, EF = DE in square ABCE (2021 Polish JMO R2 p2)
Given is the square
A
B
C
D
ABCD
A
BC
D
. Point
E
E
E
lies on the diagonal
A
C
AC
A
C
, where
A
E
>
E
C
AE> EC
A
E
>
EC
. On the side
A
B
AB
A
B
, a different point from
B
B
B
has been selected for which
E
F
=
D
E
EF = DE
EF
=
D
E
. Prove that
∠
D
E
F
=
9
0
o
\angle DEF = 90^o
∠
D
EF
=
9
0
o
.
1
1
Hide problems
b is integer if a is integer, 2a + a^2= 2b + b^2 (2021 Polish JMO R2 p1)
The numbers
a
,
b
a, b
a
,
b
satisfy the condition
2
a
+
a
2
=
2
b
+
b
2
2a + a^2= 2b + b^2
2
a
+
a
2
=
2
b
+
b
2
. Prove that if
a
a
a
is an integer,
b
b
b
is also an integer.