1996 Hungary-Israel Binational

Part of Hungary-Israel Binational

Subcontests

(4)

1

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.

1

Hungary-Israel Binational 1996\4

a_1, a_2, \cdots, a_n

is a sequence of real numbers, and

b_1, b_2, \cdots, b_n

are real numbers that satisfy the condition

1 \ge b_1 \ge b_2 \ge \cdots \ge b_n \ge 0

. Prove that there exists a natural number

k \le 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|

1

Hungary-Israel Binational 1996\2

n>2

is an integer such that

n^2

can be represented as a difference of cubes of 2 consecutive positive integers. Prove that

n

is a sum of 2 squares of positive integers, and that such

n

1

Hungary-Israel Binational 1996\1

Find all integer sequences of the form

x_i, 1 \le i \le 1997

, that satisfy \sum_{k\equal{}1}^{1997} 2^{k\minus{}1} x_{k}^{1997}\equal{}1996\prod_{k\equal{}1}^{1997}x_k.