April 2003 TRINITY MATH TRIATHLON

1.  Use the clues and the picture to determine the order of the kids:

 (a) Samuel is between two girls
 (b) Gayle is the only one wearing a hat
 (c) Two people are between Gayle and Maria
 (d) Samuel is next to Maria
 (e) Gina is on one of the ends
 (f) There is one person between Mike and Lucy.

2.  If you are given a square, you can create a rectangle by drawing a single line parallel to one of the sides of the square.  (See picture below.)  Which of the following shapes can also be made by drawing a single line across a square? Demonstrate the line drawing, if it is possible.
 

(a) Acute Triangle (b) Isosceles Triangle (c) Equilateral Triangle (d) Non-rectangular parallelogram (e) Trapezoid (f) Pentagon (g) Hexagon

3.  Divide the large square into smaller squares along the grid lines so that each square contains exactly one star.  

4.  For a pair of positive integers m and n, there is a unique smallest positive integer which can be created by taking integer combinations of m and n in the form mx + ny. 

For example, if m = 8 and n = 15, then integer combinations include 23 and 7 since 

 8 × 1 + 15 × 1 = 23 and 8 × (-1) + 15 × 1 = 7.

But the smallest integer combination is 1 since 8 × 2 + 15 × (-1) = 1.  
 

(a) Find the smallest positive integer which can be created using integer combinations of each of the following pairs of numbers.

(i)  m = 5 and n = 7        (ii)  m = 6 and n = 14
(iii)  m = 4 and n = 20   (iv)   m = 21 and n =25

(b) What familiar concept do all previous results share?

5.   A positive integer n is generated by a positive integer k if the sum of k and k 's digits is n.  Thus,  n = 25 is generated by k = 17 because 17 + 1 + 7 = 25 while n = 1 cannot be generated by any positive integer.  Find the next five integers which canNOT be generated by any other positive integer.