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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1972 Kurschak Competition
1972 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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for some points P_i it is not possible with a total length less than 25
A
B
C
D
ABCD
A
BC
D
is a square side
10
10
10
. There are four points
P
1
,
P
2
,
P
3
,
P
4
P_1, P_2, P_3, P_4
P
1
,
P
2
,
P
3
,
P
4
inside the square. Show that we can always construct line segments parallel to the sides of the square of total length
25
25
25
or less, so that each
P
i
P_i
P
i
is linked by the segments to both of the sides
A
B
AB
A
B
and
C
D
CD
C
D
. Show that for some points
P
i
P_i
P
i
it is not possible with a total length less than
25
25
25
.
2
1
Hide problems
f(X)$ be the number of ways of dividing the line of n boys and n girls
A class has
n
>
1
n > 1
n
>
1
boys and
n
n
n
girls. For each arrangement
X
X
X
of the class in a line let
f
(
X
)
f(X)
f
(
X
)
be the number of ways of dividing the line into two non-empty segments, so that in each segment the number of boys and girls is equal. Let the number of arrangements with
f
(
X
)
=
0
f(X) = 0
f
(
X
)
=
0
be
A
A
A
, and the number of arrangements with
f
(
X
)
=
1
f(X) = 1
f
(
X
)
=
1
be
B
B
B
. Show that
B
=
2
A
B = 2A
B
=
2
A
.
1
1
Hide problems
a(b -c)^2 + b(c - a)^2 + c(a - b)^2 + 4abc > a^3 + b^3 + c^3
A triangle has side lengths
a
,
b
,
c
a, b, c
a
,
b
,
c
. Prove that
a
(
b
−
c
)
2
+
b
(
c
−
a
)
2
+
c
(
a
−
b
)
2
+
4
a
b
c
>
a
3
+
b
3
+
c
3
a(b -c)^2 + b(c - a)^2 + c(a - b)^2 + 4abc > a^3 + b^3 + c^3
a
(
b
−
c
)
2
+
b
(
c
−
a
)
2
+
c
(
a
−
b
)
2
+
4
ab
c
>
a
3
+
b
3
+
c
3