**USA**

**Mathematical Talent Search**

**PROBLEMS**

**Round 4 - Year 13 - Academic Year 2001-2002**

**1/4/13.**In a strange language there are only two letters,

*a*and

*b*, and it is postulated that the letter

*a*is a word. Furthermore, all additional words are formed according to the following rules:

A. Given any word, a new word can
be formed from it by adding a

*b*at the righthand end.
B. If in any word a sequence

*aaa*appears, a new word can be formed by replacing the*aaa*by the letter*b*.
C. If in any word the sequence

*bbb*appears, a new word can be formed by omitting*bbb*.
D. Given any word, a new word can be formed by writing down the
sequence that constitutes the given word twice.

For example, by (D),

*aa*is a word, and by (D) again,*aaaa*is a word. Hence by (B)*ba*is a word, and by (A)*bab*is also a word. Again, by (A),*babb*is a word, and so by (D),*babbbabb*is also a word. Finally, by (C) we find that*baabb*is a word.
Prove that in this language

*baabaabaa*is not a word.**2/4/13.**Let

*f (x) = x .*⎿

*x*.⎿

*x*. ⎿

*x*⏌⏌⏌

*for all positive real numbers*

*x*, where ⎿

*y*⏌

*denotes the greatest integer less than or equal to*

*y*.

(1)
Determine

*x*so that*f(x) =*2001
(2)
Prove that

*f(x) =*2002 has no solution.**3/4/13.**Let f be a function defined on the set of all integers, and assume that it satisfies the following properties:

A.

*f*(0) ≠ 0;
B.

*f*(1) = 3; and
C.

*f(x*)*f(y)*=*f(x*+*y*) +*f(x*–*y*) for all integers x and y.
Determine

*f*(7) .**4/4/13.**A certain company has a faulty telephone system that sometimes transposes a pair of adjacent digits when someone dials a three-digit extension. Hence a call to

*x*318 would ring at either

*x*318,

*x*138, or

*x*381 , while a call received at

*x*044 would be intended for either

*x*404 or

*x*044 . Rather than replace the system, the company is adding a computer to deduce which dialed extensions are in error and revert those numbers to their correct form. They have

to leave out several possible extensions for this to work. What is
the greatest number of three-digit extensions the company can assign under this
plan?

**5/4/13.**Determine the smallest number of squares into which one can dissect a 11 × 13 rectangle, and exhibit such a dissection. The squares need not be of different sizes, their bases should be integers, and they should not overlap.

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