SETON HALL UNIVERSITY

 

EIGHTEENTH ANNUAL

 

JOSEPH W. ANDRUSHKIW

 

MATHEMATICS COMPETITION

 

 

1. The sum of the ages of Al and Bob now is twice the sum of their ages 5 years ago; Al is older than Bob. Bob’s age now exceeds the difference in their ages (now) by 1 year. Find Bob’s age now.

 

 

2. Find the difference in the degree measure of each angle of a regular polygon having 16 sides and in each angle of a regular octagon.

 

 

3. How many distinguishable arrangements are there of 4 identical red blocks and 3 identical green blocks in a row with the same color block at each end?        

 

 

 4. Let M equal the sum of the reciprocals of the squares of the four smallest positive integers, and let N be the square of the product of the four largest negative integers. Find the value of MN.

                                               

 

5. The real values of x for which  form an interval. Find the length of this interval.

 

 

6. Solve for x in terms of r (where x and r are positive real numbers, r > 1): .

 

 

7. A swimming pool can be filled using pipe A or pipe B (or both) and emptied using pipe C. If pipes A and B are both open (and pipe C is closed) the pool can be filled in 1 hours. If pipes A and B are open (and C closed) for 1 hour, and then pipe A is closed, B is left open and C is opened, the total time to fill the pool is 2 hours. If it takes 3 times as long to empty the pool using pipe C (with pipes A and B closed) as it does to fill the pool using pipe B (with pipes A and C closed), how long does it take to fill the pool using pipe A alone (with pipes B and C closed)?

 

 

8. Simplify (where w is a real number, w > 1): .

 

 

 

9. Quadrilateral ABCD has right angles at C and D; angle B has measure ; sides AB and BC are 6 yards and 10 yards long, respectively. How much shorter is side AD than diagonal AC?

 

 

 

10. The equation  has roots  (where A, B, C, D, E, F are real numbers). Given that A = 20, find the value of  

 

 

 

11. The product of 4 consecutive even positive integers exceeds 7350 times the sum of the 4 integers by 2520. Find the sum of the squares of these 4 integers.

 

 

 

12. Write in simplest form in terms of sin A : .

 

 

 

13. Jessica has coupons worth $1, $2, $5, and $10 each. She has a total of 80 coupons of total value $250. The number of $1 and $5 coupons she has exceeds the number of $2 and $10 coupons by 12. Had she half as many $1 coupons, double the number of $2 coupons, one-third the number of $5 coupons and three times as many $10 coupons, the value of the coupons would be $459. Find the value of the $5 coupons which Jessica has.

 

 

 

14. Four integers from the set H = {1,2,3, … ,20} are chosen at random, without replacement, and arranged in order of size (smallest to largest). Find the probability that the four numbers chosen (after rearrangement) form an arithmetic progression.

 

 

 

15. The point P = (8,-2) and the circle with equation  lie on a coordinate plane. Line L₁ through P meets the circle in one point Q = (2,6). Line L₂ through P meets the circle in points R = (2,0) and S. An equation of the line that passes through points Q and S can be written in the form , where m and b are real numbers. Find the value of 

 

 

 

16. Find the value of  in exact rational form, given that  and .