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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2021 Kurschak Competition
2021 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
Hide problems
Coincidence of diagonals in hexagon
Let
A
1
B
3
A
2
B
1
A
3
B
2
A_1B_3A_2B_1A_3B_2
A
1
B
3
A
2
B
1
A
3
B
2
be a cyclic hexagon such that
A
1
B
1
,
A
2
B
2
,
A
3
B
3
A_1B_1,A_2B_2,A_3B_3
A
1
B
1
,
A
2
B
2
,
A
3
B
3
intersect at one point. Let
C
1
=
A
1
B
1
∩
A
2
A
3
,
C
2
=
A
2
B
2
∩
A
1
A
3
,
C
3
=
A
3
B
3
∩
A
1
A
2
C_1=A_1B_1\cap A_2A_3,C_2=A_2B_2\cap A_1A_3,C_3=A_3B_3\cap A_1A_2
C
1
=
A
1
B
1
∩
A
2
A
3
,
C
2
=
A
2
B
2
∩
A
1
A
3
,
C
3
=
A
3
B
3
∩
A
1
A
2
. Let
D
1
D_1
D
1
be the point on the circumcircle of the hexagon such that
C
1
B
1
D
1
C_1B_1D_1
C
1
B
1
D
1
touches
A
2
A
3
A_2A_3
A
2
A
3
. Define
D
2
,
D
3
D_2,D_3
D
2
,
D
3
analogously. Show that
A
1
D
1
,
A
2
D
2
,
A
3
D
3
A_1D_1,A_2D_2,A_3D_3
A
1
D
1
,
A
2
D
2
,
A
3
D
3
meet at one point.
2
1
Hide problems
Odd cycles in graph
In neverland, there are
n
n
n
cities and
n
n
n
airlines. Each airline serves an odd number of cities in a circular way, that is, if it serves cities
c
1
,
c
2
,
…
,
c
2
k
+
1
c_1,c_2,\dots,c_{2k+1}
c
1
,
c
2
,
…
,
c
2
k
+
1
, then they fly planes connecting
c
1
c
2
,
c
2
c
3
,
…
,
c
1
c
2
k
+
1
c_1c_2,c_2c_3,\dots,c_1c_{2k+1}
c
1
c
2
,
c
2
c
3
,
…
,
c
1
c
2
k
+
1
. Show that we can select an odd number of cities
d
1
,
d
2
,
…
,
d
2
m
+
1
d_1,d_2,\dots,d_{2m+1}
d
1
,
d
2
,
…
,
d
2
m
+
1
such that we can fly
d
1
→
d
2
→
⋯
→
d
2
m
+
1
→
d
1
d_1\rightarrow d_2\rightarrow\dots\rightarrow d_{2m+1}\rightarrow d_1
d
1
→
d
2
→
⋯
→
d
2
m
+
1
→
d
1
while using each airline at most once.
1
1
Hide problems
Areas forming geometric sequence
Let
P
0
=
(
a
0
,
b
0
)
,
P
1
=
(
a
1
,
b
1
)
,
P
2
=
(
a
2
,
b
2
)
P_0=(a_0,b_0),P_1=(a_1,b_1),P_2=(a_2,b_2)
P
0
=
(
a
0
,
b
0
)
,
P
1
=
(
a
1
,
b
1
)
,
P
2
=
(
a
2
,
b
2
)
be points on the plane such that
P
0
P
1
P
2
Δ
P_0P_1P_2\Delta
P
0
P
1
P
2
Δ
contains the origin
O
O
O
. Show that the areas of triangles
P
0
O
P
1
,
P
0
O
P
2
,
P
1
O
P
2
P_0OP_1,P_0OP_2,P_1OP_2
P
0
O
P
1
,
P
0
O
P
2
,
P
1
O
P
2
form a geometric sequence in that order if and only if there exists a real number
x
x
x
, such that
a
0
x
2
+
a
1
x
+
a
2
=
b
0
x
2
+
b
1
x
+
b
2
=
0
a_0x^2+a_1x+a_2=b_0x^2+b_1x+b_2=0
a
0
x
2
+
a
1
x
+
a
2
=
b
0
x
2
+
b
1
x
+
b
2
=
0