Math Problems From USA Math Talent Search

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 x318 would ring at either x318, x138, or x381 , while a call received at x044 would be intended for either x404 or x044 . 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.