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Hungary-Israel Binational
1996 Hungary-Israel Binational
1996 Hungary-Israel Binational
Part of
Hungary-Israel Binational
Subcontests
(4)
3
1
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Hungary-Israel Binational 1996\3
A given convex polyhedron has no vertex which belongs to exactly 3 edges. Prove that the number of faces of the polyhedron that are triangles, is at least 8.
4
1
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Hungary-Israel Binational 1996\4
a
1
,
a
2
,
⋯
,
a
n
a_1, a_2, \cdots, a_n
a
1
,
a
2
,
⋯
,
a
n
is a sequence of real numbers, and
b
1
,
b
2
,
⋯
,
b
n
b_1, b_2, \cdots, b_n
b
1
,
b
2
,
⋯
,
b
n
are real numbers that satisfy the condition
1
≥
b
1
≥
b
2
≥
⋯
≥
b
n
≥
0
1 \ge b_1 \ge b_2 \ge \cdots \ge b_n \ge 0
1
≥
b
1
≥
b
2
≥
⋯
≥
b
n
≥
0
. Prove that there exists a natural number
k
≤
n
k \le n
k
≤
n
that satisifes |a_1b_1 \plus{} a_2b_2 \plus{} \cdots \plus{} a_nb_n| \le |a_1 \plus{} a_2 \plus{} \cdots \plus{} a_k|
2
1
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Hungary-Israel Binational 1996\2
n
>
2
n>2
n
>
2
is an integer such that
n
2
n^2
n
2
can be represented as a difference of cubes of 2 consecutive positive integers. Prove that
n
n
n
is a sum of 2 squares of positive integers, and that such
n
n
n
does exist.
1
1
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Hungary-Israel Binational 1996\1
Find all integer sequences of the form
x
i
,
1
≤
i
≤
1997
x_i, 1 \le i \le 1997
x
i
,
1
≤
i
≤
1997
, that satisfy \sum_{k\equal{}1}^{1997} 2^{k\minus{}1} x_{k}^{1997}\equal{}1996\prod_{k\equal{}1}^{1997}x_k.