▪ Puerto Rico Team Selection Test 2012

1. Let and be consecutive integers such that

Find the maximum value of . 

2. A cone is constructed with a semicircular piece of paper, with radius 10. Find the height of the cone. 

3. is a triangle that is inscribed in a circle. The angle bisectors of meet the circle at , respectively. Show that is perpendicular to . 

4. Let be digits such that . How many numbers of the form are

multiples of ? 

5. A point is outside of a circle and the distance to the center is . A secant line from meets the circle at and so that the exterior segment of the secant, , is and is . Find the radius of the circle.
 6. The increasing sequence is formed with positive integers which are powers of or sums of different powers of . Which number is in the position? 

7. Let be a function with the following properties:

1) is defined for every positive integer ;

2) is an integer;

3) ;

4) for all and ;

5) whenever . Prove that .

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