2008 Hungary-Israel Binational

Part of Hungary-Israel Binational

Subcontests

(3)

2

Hungary-Israel Binational 2008\2

For every natural number

t

f(t)

is the probability that if a fair coin is tossed

t

times, the number of times we get heads is 2008 more than the number of tails. What is the value of

t

f(t)

attains its maximum? (if there is more than one, describe all of them)

Hungary-Israel Binational 2008\5

a_n

is defined as follows: a_0\equal{}1, a_1\equal{}1, a_{n\plus{}1}\equal{}\frac{1\plus{}a_{n}^2}{a_{n\minus{}1}}. Prove that all the terms of the sequence are integers.

2

Hungary-Israel Binational 2008\1

Find the largest value of n, such that there exists a polygon with n sides, 2 adjacent sides of length 1, and all his diagonals have an integer length.

Hungary-Israel Binational 2008\4

Prove that: \sum_{i\equal{}1}^{n^2} \lfloor \frac{i}{3} \rfloor\equal{} \frac{n^2(n^2\minus{}1)}{6} For all

n \in N

2

Special partition of a rectangle in smaller rectangles.

D

is partitioned in several (

\ge2

) rectangles with sides parallel to those of

D

. Given that any line parallel to one of the sides of

D

, and having common points with the interior of

D

, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with

D

's boundary.
Author: Kei Irie, Japan

Hungary-Israel Binational 2008\6

P and Q are 2 points in the area bounded by 2 rays, e and f, coming out from a point O. Describe how to construct, with a ruler and a compass only, an isosceles triangle ABC, such that his base AB is on the ray e, the point C is on the ray f, P is on AC, and Q on BC.