1999 Hungary-Israel Binational

Part of Hungary-Israel Binational

Subcontests

(3)

2

Hungary-Israel Binational 1999\2

2n\plus{}1 lines are drawn in the plane, in such a way that every 3 lines define a triangle with no right angles. What is the maximal possible number of acute triangles that can be made in this way?

Hungary-Israel Binational 1999\5

The function f(x,y,z)\equal{}\frac{x^2\plus{}y^2\plus{}z^2}{x\plus{}y\plus{}z} is defined for every

x,y,z \in R

whose sum is not 0. Find a point

(x_0,y_0,z_0)

such that 0 < x_0^2\plus{}y_0^2\plus{}z_0^2 < \frac{1}{1999} and

1.999 < f(x_0,y_0,z_0) < 2

2

Hungary-Israel Binational 1999\3

Find all functions

f:\mathbb{Q}\to\mathbb{R}

that satisfy f(x\plus{}y)\equal{}f(x)f(y)\minus{}f(xy)\plus{}1 for every

x,y\in\mathbb{Q}

Hungary-Israel Binational 1999\6

In a multiple-choice test, there are 4 problems, each having 3 possible answers. In some group of examinees, it turned out that for every 3 of them, there was a question that each of them gave a different answer to. What is the maximal number of examinees in this group?

2

Hungary-Israel Binational 1999\1

f(x)

is a given polynomial whose degree at least 2. Define the following polynomial-sequence: g_1(x)\equal{}f(x), g_{n\plus{}1}(x)\equal{}f(g_n(x)), for all

n \in N

r_n

be the average of

g_n(x)

's roots. If r_{19}\equal{}99, find

r_{99}

Hungary-Israel Binational 1999\4

c

is a positive integer. Consider the following recursive sequence: a_1\equal{}c, a_{n\plus{}1}\equal{}ca_{n}\plus{}\sqrt{(c^2\minus{}1)(a_n^2\minus{}1)}, for all

n \in N

. Prove that all the terms of the sequence are positive integers.