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Contests
International Contests
Hungary-Israel Binational
1991 Hungary-Israel Binational
1991 Hungary-Israel Binational
Part of
Hungary-Israel Binational
Subcontests
(4)
4
1
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Hungary-Israel Binational 1991_4
Find all the real values of
λ
\lambda
λ
for which the system of equations x\plus{}y\plus{}z\plus{}v\equal{}0 and \left(xy\plus{}yz\plus{}zv\right)\plus{}\lambda\left(xz\plus{}xv\plus{}yv\right)\equal{}0, has a unique real solution.
3
1
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Hungary-Israel Binational 1991_3
Let
H
n
\mathcal{H}_n
H
n
be the set of all numbers of the form
2
±
2
±
2
±
…
±
2
2 \pm\sqrt{2 \pm\sqrt{2 \pm\ldots\pm\sqrt 2}}
2
±
2
±
2
±
…
±
2
where "root signs" appear
n
n
n
times. (a) Prove that all the elements of
H
n
\mathcal{H}_n
H
n
are real. (b) Computer the product of the elements of
H
n
\mathcal{H}_n
H
n
. (c) The elements of
H
11
\mathcal{H}_{11}
H
11
are arranged in a row, and are sorted by size in an ascending order. Find the position in that row, of the elements of
H
11
\mathcal{H}_{11}
H
11
that corresponds to the following combination of
±
\pm
±
signs: \plus{}\plus{}\plus{}\plus{}\plus{}\minus{}\plus{}\plus{}\minus{}\plus{}\minus{}
2
1
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Hungary-Israel Binational 1991_2
The vertices of a square sheet of paper are
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
. The sheet is folded in a way that the point
D
D
D
is mapped to the point
D
′
D'
D
′
on the side
B
C
BC
BC
. Let
A
′
A'
A
′
be the image of
A
A
A
after the folding, and let
E
E
E
be the intersection point of
A
B
AB
A
B
and
A
′
D
′
A'D'
A
′
D
′
. Let
r
r
r
be the inradius of the triangle
E
B
D
′
EBD'
EB
D
′
. Prove that r\equal{}A'E.
1
1
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Polynomial
Suppose
f
(
x
)
f(x)
f
(
x
)
is a polynomial with integer coefficients such that
f
(
0
)
=
11
f(0) = 11
f
(
0
)
=
11
and
f
(
x
1
)
=
f
(
x
2
)
=
.
.
.
=
f
(
x
n
)
=
2002
f(x_1) = f(x_2) = ... = f(x_n) = 2002
f
(
x
1
)
=
f
(
x
2
)
=
...
=
f
(
x
n
)
=
2002
for some distinct integers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, . . . , x_n
x
1
,
x
2
,
...
,
x
n
. Find the largest possible value of
n
n
n
.