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

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 points)

**B. 3950.** Let *H* be the set of integers not greater than 2006: *H*={1,2,...,2006}. Let *D *denote the number of those subsets of the set *H* in which the number of elements divided by 32 leaves a remainder of 7, and let *S* denote the number of those subsets of the set *H *in which the number of elements divided by 16 leaves a remainder of 14. Prove that *S*=2*D*.

(Based on an OKTV (National Competition) problem)

(5 points)

**K. 97.** There are 7 houses in Dragon Street, numbered 1 to 7. Last week, Mr. D. E. Livery the postman delivered letters to two houses every day. There are two houses where he did not brought anything during the whole week, but he visited all the other houses twice. The sum of the number of the houses visited by Mr. Livery was 5 on Monday, 8 on Tuesday, 9 on Wednesday, 13 on Thursday and 7 on Friday. (There is no mail on Saturday and Sunday.) Which are the two houses that did not receive any letter last week?

(6 points)

This problem is for grade 9 students only.