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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2019 Kurschak Competition
2019 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
1
1
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Circle passing through the midpoint of BC
In an acute triangle
△
A
B
C
\bigtriangleup ABC
△
A
BC
,
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
, and
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
are the projections of
A
,
B
,
C
A,B,C
A
,
B
,
C
to the corresponding sides. Let the reflection of
B
1
B_1
B
1
wrt
C
C
1
CC_1
C
C
1
be
Q
Q
Q
, and the reflection of
C
1
C_1
C
1
wrt
B
B
1
BB_1
B
B
1
be
P
P
P
. Prove that the circumcirle of
A
1
P
Q
A_1PQ
A
1
PQ
passes through the midpoint of
B
C
BC
BC
.
2
1
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Family of sets with a regular property
Find all family
F
\mathcal{F}
F
of subsets of
[
n
]
[n]
[
n
]
such that for any nonempty subset
X
⊆
[
n
]
X\subseteq [n]
X
⊆
[
n
]
, exactly half of the elements
A
∈
F
A\in \mathcal{F}
A
∈
F
satisfies that
∣
A
∩
X
∣
|A\cap X|
∣
A
∩
X
∣
is even.
3
1
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Sum of sets
Is it true that if
H
H
H
and
A
A
A
are bounded subsets of
R
\mathbb{R}
R
, then there exists at most one set
B
B
B
such that
a
+
b
(
a
∈
A
,
b
∈
B
)
a+b(a\in A,b\in B)
a
+
b
(
a
∈
A
,
b
∈
B
)
are pairwise distinct and
H
=
A
+
B
H=A+B
H
=
A
+
B
.