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Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1977 Kurschak Competition
1977 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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each student knows a total of n + 1 students at the other two schools
Three schools each have
n
n
n
students. Each student knows a total of
n
+
1
n + 1
n
+
1
students at the other two schools. Show that there must be three students, one from each school, who know each other.
2
1
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concyclic wanted
A
B
C
ABC
A
BC
is a triangle with orthocenter
H
H
H
. The median from
A
A
A
meets the circumcircle again at
A
1
A_1
A
1
, and
A
2
A_2
A
2
is the reflection of
A
1
A_1
A
1
in the midpoint of
B
C
BC
BC
. The points
B
2
B_2
B
2
and
C
2
C_2
C
2
are defined similarly. Show that
H
H
H
,
A
2
A_2
A
2
,
B
2
B_2
B
2
and
C
2
C_2
C
2
lie on a circle. https://cdn.artofproblemsolving.com/attachments/f/1/192d14a0a7a9aa9ac7b38763e6ea6a4a95be55.png
1
1
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n^4 + 4^n never prime greater than 5
Show that there are no integers
n
n
n
such that
n
4
+
4
n
n^4 + 4^n
n
4
+
4
n
is a prime greater than
5
5
5
.