Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation

Already signed up?
New to KöMaL?

Problem A. 395. (March 2006)

A. 395. Let 1<a<2 be a real number. (a) Show that there exists a unique sequence x₁,x₂,... of positive integers satisfying x_i+1 $\ge$ x_i² for all indices i and

$\left(1+\frac1{x_1}\right)\left(1+\frac1{x_2}\right)\dots = a.$

(b) Prove that inequality x_i+1>x_i² holds for infinitely many indices if and only if a is irrational.

(5 pont)

Deadline expired on April 18, 2006.

Statistics:

11 students sent a solution.

5 points: Hujter Bálint, Kisfaludi-Bak Sándor, Nagy 224 Csaba, Paulin Roland.

4 points: Erdélyi Márton, Tomon István.

3 points: 1 student.

2 points: 3 students.

1 point: 1 student.

Problems in Mathematics of KöMaL, March 2006

11 students sent a solution.
5 points:	Hujter Bálint, Kisfaludi-Bak Sándor, Nagy 224 Csaba, Paulin Roland.
4 points:	Erdélyi Márton, Tomon István.
3 points:	1 student.
2 points:	3 students.
1 point:	1 student.