MathDB
Problems
Contests
International Contests
Hungary-Israel Binational
2006 Hungary-Israel Binational
2006 Hungary-Israel Binational
Part of
Hungary-Israel Binational
Subcontests
(3)
3
2
Hide problems
Hungary-Israel Binational 2006_3
Let \mathcal{H} \equal{} A_1A_2\ldots A_n be a convex
n
n
n
-gon. For i \equal{} 1, 2, \ldots, n, let
A
i
′
A'_{i}
A
i
′
be the point symmetric to
A
i
A_i
A
i
with respect to the midpoint of A_{i \minus{} 1}A_{i \plus{} 1} (where A_{n \plus{} 1} \equal{} A_1). We say that the vertex
A
i
A_i
A
i
is good if
A
i
′
A'_{i}
A
i
′
lies inside
H
\mathcal{H}
H
. Show that at least n \minus{} 3 vertices of
H
\mathcal{H}
H
are good.
Hungary-Israel Binational 2006_6
A group of
100
100
100
students numbered
1
1
1
through
100
100
100
are playing the following game. The judge writes the numbers
1
1
1
,
2
2
2
,
…
\ldots
…
,
100
100
100
on
100
100
100
cards, places them on the table in an arbitrary order and turns them over. The students
1
1
1
to
100
100
100
enter the room one by one, and each of them flips
50
50
50
of the cards. If among the cards flipped by student
j
j
j
there is card
j
j
j
, he gains one point. The flipped cards are then turned over again. The students cannot communicate during the game nor can they see the cards flipped by other students. The group wins the game if each student gains a point. Is there a strategy giving the group more than
1
1
1
percent of chance to win?
2
2
Hide problems
Hungary-Israel Binational 2006_2
A block of size
a
×
b
×
c
a\times b\times c
a
×
b
×
c
is composed of
1
×
1
×
2
1\times 1\times 2
1
×
1
×
2
domino blocks. Assuming that each of the three possible directions of domino blocks occurs equally many times, what are the possible values of
a
a
a
,
b
b
b
,
c
c
c
?
Hungary-Israel Binational 2006_5
If
x
x
x
,
y
y
y
,
z
z
z
are nonnegative real numbers with the sum
1
1
1
, find the maximum value of S \equal{} x^2(y \plus{} z) \plus{} y^2(z \plus{} x) \plus{} z^2(x \plus{} y) and C \equal{} x^2y \plus{} y^2z \plus{} z^2x.
1
2
Hide problems
Hungary-Israel Binational 2006_1
If natural numbers
x
x
x
,
y
y
y
,
p
p
p
,
n
n
n
,
k
k
k
with
n
>
1
n > 1
n
>
1
odd and
p
p
p
an odd prime satisfy x^n \plus{} y^n \equal{} p^k, prove that
n
n
n
is a power of
p
p
p
.
Hungary-Israel Binational 2006_4
A point
P
P
P
inside a circle is such that there are three chords of the same length passing through
P
P
P
. Prove that
P
P
P
is the center of the circle.