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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1950 Kurschak Competition
1950 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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triples of real numbers
(
x
1
,
y
1
,
z
1
)
(x_1, y_1,z_1)
(
x
1
,
y
1
,
z
1
)
and
(
x
2
,
y
2
,
z
2
)
(x_2, y_2, z_2)
(
x
2
,
y
2
,
z
2
)
are triples of real numbers such that for every pair of integers
(
m
,
n
)
(m,n)
(
m
,
n
)
at least one of
x
1
m
+
y
1
n
+
z
1
x_{1m} + y_{1n} + z_1
x
1
m
+
y
1
n
+
z
1
,
x
2
m
+
y
2
n
+
z
2
x_{2m} + y_{2n} + z_2
x
2
m
+
y
2
n
+
z
2
is an even integer. Prove that one of the triples consists of three integers.
2
1
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concurrency wanted, 3 tangent circles
Three circles
C
1
C_1
C
1
,
C
2
C_2
C
2
,
C
3
C_3
C
3
in the plane touch each other (in three different points). Connect the common point of
C
1
C_1
C
1
and
C
2
C_2
C
2
with the other two common points by straight lines. Show that these lines meet
C
3
C_3
C
3
in diametrically opposite points.
1
1
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several people visited a library yesterday
Several people visited a library yesterday. Each one visited the library just once (in the course of yesterday). Amongst any three of them, there were two who met in the library. Prove that there were two moments
T
T
T
and
T
T
T
' yesterday such that everyone who visited the library yesterday was in the library at
T
T
T
or
T
′
T'
T
′
(or both).