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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1964 Kurschak Competition
1964 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
Hide problems
\sqrt{(w^2 + x^2 + y^2 + z^2)/4} >=\sqrt[3]{wxy + wxz + wyz + xyz)/4}
Show that for any positive reals
w
,
x
,
y
,
z
w, x, y, z
w
,
x
,
y
,
z
we have
w
2
+
x
2
+
y
2
+
z
2
4
≥
w
x
y
+
w
x
z
+
w
y
z
+
x
y
z
4
3
\sqrt{\frac{w^2 + x^2 + y^2 + z^2}{4}}\ge \sqrt[3]{ \frac{wxy + wxz + wyz + xyz}{4}}
4
w
2
+
x
2
+
y
2
+
z
2
≥
3
4
w
x
y
+
w
x
z
+
w
yz
+
x
yz
2
1
Hide problems
dances of boys and girls at a party
At a party every girl danced with at least one boy, but not with all of them. Similarly, every boy danced with at least one girl, but not with all of them. Show that there were two girls
G
G
G
and
G
′
G'
G
′
and two boys
B
B
B
and
B
′
B'
B
′
, such that each of
B
B
B
and
G
G
G
danced,
B
′
B'
B
′
and
G
′
G'
G
′
danced, but
B
B
B
and
G
′
G'
G
′
did not dance, and
B
′
B'
B
′
and
G
G
G
did not dance.
1
1
Hide problems
ratio DD'/AB wanted, ABCD, ABCD' congruent tetrahedra
A
B
C
ABC
A
BC
is an equilateral triangle.
D
D
D
and
D
′
D'
D
′
are points on opposite sides of the plane
A
B
C
ABC
A
BC
such that the two tetrahedra
A
B
C
D
ABCD
A
BC
D
and
A
B
C
D
′
ABCD'
A
BC
D
′
are congruent (but not necessarily with the vertices in that order). If the polyhedron with the five vertices
A
,
B
,
C
,
D
,
D
′
A, B, C, D, D'
A
,
B
,
C
,
D
,
D
′
is such that the angle between any two adjacent faces is the same, find
D
D
′
/
A
B
DD'/AB
D
D
′
/
A
B
.