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National and Regional Contests
Hungary Contests
Kürschák Math Competition
1957 Kurschak Competition
1957 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
2
1
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several types of mug, each with 2 colors, chosen from a set of 6
A factory produces several types of mug, each with two colors, chosen from a set of six. Every color occurs in at least three different types of mug. Show that we can find three mugs which together contain all six colors.
3
1
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max of |a_1 - 1| + |a_2-2|+...+ |a_n- n| where a_i is permutation of i
What is the largest possible value of
∣
a
1
−
1
∣
+
∣
a
2
−
2
∣
+
.
.
.
+
∣
a
n
−
n
∣
|a_1 - 1| + |a_2-2|+...+ |a_n- n|
∣
a
1
−
1∣
+
∣
a
2
−
2∣
+
...
+
∣
a
n
−
n
∣
where
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2,..., a_n
a
1
,
a
2
,
...
,
a
n
is a permutation of
1
,
2
,
.
.
.
,
n
1,2,..., n
1
,
2
,
...
,
n
?
1
1
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locin in 3d, tetrahedron related
A
B
C
ABC
A
BC
is an acute-angled triangle.
D
D
D
is a variable point in space such that all faces of the tetrahedron
A
B
C
D
ABCD
A
BC
D
are acute-angled.
P
P
P
is the foot of the perpendicular from
D
D
D
to the plane
A
B
C
ABC
A
BC
. Find the locus of
P
P
P
as
D
D
D
varies.