Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem A. 412. (November 2006)

A. 412. Let t be an irrational number and let


f(x,y) = \cases{
1, & \text{if \ } \{tx\}>\{ty\}; \\
0, & \text{if \ }\{tx\}\le\{ty\}.
}

for all integers x, y. The function f(x,y) has many decompositions of the form f(x,y)=u(x)+v(y)+w(x-y), where the functions u, v and w map the set of integers to the set of reals. Prove that (a) There exists a case when u, v and w are bounded; (b) There exists a case when u, v and w attain only integers; (c) There exists no case when u, v and w are bounded and attain only integers.

(Proposed by Tamás Keleti, Budapest)

(5 pont)

Deadline expired on December 15, 2006.


Statistics:

10 students sent a solution.
5 points:Gyenizse Gergő, Hujter Bálint, Kisfaludi-Bak Sándor, Korándi Dániel, Lovász László Miklós, Nagy 224 Csaba, Nagy 235 János, Tomon István.
3 points:2 students.

Problems in Mathematics of KöMaL, November 2006