Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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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 xi+1\gexi2 for all indices i and


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

(b) Prove that inequality xi+1>xi2 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