Download 1 - Grissom Math Team

Grissom Math Tournament
Comprehensive Test
April 11, 2009
1. If G = the sum of the first two even composite numbers, H= the sum of the first 3 prime numbers, and
S = the product of the first three positive non-prime integers, find GHS.
A.1920
B. 2400
C. 3600
D. 7200
E. 19200
2. If Louise can sharpen a pencil in three minutes, Al can sharpen 3 pencils in a minute, and George can
sharpen 3 pencils in 3 minutes, how many minutes would it take them to sharpen 143 pencils, working
together? Assume all the pencils are identical, and everyone works at contstant rates.
A. 33
B.
22
7
C. 27
D. 11
3. Which has the greatest area, an equilateral triangle with side length
regular hexagon with side length
A. triangle
B. square
E. 32
1
1
, a square with side length , a
4
3
1
1
, or a circle with radius ?
6
7
C. hexagon
4. If the sum of the roots of the equation x 2  9x  3  0 is
D. circle
E. NOTA
b
and the product of the roots of the same
a
b
a
equation is   a , find a+b.
A. 48
B. 40
C. 1
D. -96
E. -1
5. Find the minimum possible value of the sum of an infinite geometric series with a positive
1
.
3
4
C.
3
common ratio and with a second term of
A.
3
2
B.
2
3
D.
5
4
E. NOTA
6. Casey’s right hand challenges her left hand to a thumb war, and because of her amazing flexibility this war
is actually possible. If Casey eats her Wheaties the morning of the war, the probability that her right hand
1
1
, however, if she does not eat her Wheaties, the probability that her left hand wins is . If the
5
5
1
probability that her right hand and her left hand tie is always and the probability that she eats her
5
1
Wheaties on any given morning is , what is the probability that her right hand wins the war?
5
2
13
14
16
3
A.
B.
C.
D.
E.
5
5
25
25
25
wins is
7. Evaluate: (2  i) 5  (2  i) 5
A. -76
B. -38
C. 19
D. -19
E. 0
1
8. If f (x)  x  2 and g(x)  4x  8 , find (f (g(g(f (f (2))))))) .
4
B. 
A. -2
3
2
C. 0
D. 1
E. 2
9. If the units digit of the square of an integer, x, is 4, which of the following could the units digit of the
square of (x+1)?
A. 8
10. Evaluate:
B. 6
D. 4
E. 1
C. 8
D. 10
E. NOTA
4
15 
4
15 
A. 4
C. 5
15  ...
B. 6
11. In equilateral triangle ABC, D, E, and F are points on AB , BC , and CA , respectively. If AD=2BD,
BE=2CE, CF=2AF, and the area of triangle ABC is 27, what is the area of triangle DEF?
A. 6 3
B.
13 3
2
C.
45 3
4
D. 6 6
E. NOTA
12. A total of 27 cubes of side length 1 are put together to form a larger cube of side length 3. Dave starts on a
vertex of this larger cube and wants to go to Nikhil’s house on the opposite vertex of the larger cube for a
cup of tea. A “move” is defined as moving a distance of exactly 1 along any of the edges of one of the
original 1x1x1 cubes without getting closer to Dave’s starting point. If a “path” is defined as a sequence of
9 consecutive “moves,” how many different “paths” could Dave take from his starting point to Nikhil’s
house?
A. 420
B. 840
13. Find the hundreds digit of 201
A. 0
B. 2
201
C. 864
D. 1600
E. 1680
C. 4
D. 6
E. 8
.
14. Find the coefficient of the a 2 d 2 f term in the expansion of (a  2b  3c  4d  5e  6 f ) 5 .
A. 2880
B. 1680
C. 1080
D. 480
E. -1080
15. How many positive integers less than 1000 have exactly 5 positive integral factors?
A. 0
B. 4
C. 5
D. 8
 2  3 4
16. Find the determinant of the inverse of the following matrix. 1 0 3
5 1 0
1
1
A. 
B. -35
C. -47
D. 
35
47
E. 10
E. NOTA
17. Find the area of a regular polygon with 12 sides and side length 1.
A. 6  3 3
B. 4  3
C. 8  2 3
D. 2  3
E. NOTA
18. A “boring word” is defined as any sequence of letters in which no consonant is preceded by or followed by
another consonant. What is the probability that a random arrangement of the letters in “various” (each
possible arrangement being equally likely) will be a “boring word?”
A.
14
35
B.
3
7
C.
16
35
D.
17
35
E.
18
35
19. Find the sum of the squares of the values of θ between 0 and 2π that satisfy the following equation:
4 tan  sin 4 
 sin(2 )  2 cos 
2 cos   sin(2 )
A. 4π
B.
40
π
9
C. 6π
D.
44
π
9
E.
21
π
4
20. Find the sum of the digits of x, given that x is a positive integral solution to the equation x
A. 16
B. 18
C. 20
D. 22
2
3
x
1
2
 108 .
E. 24
21. Evaluate: sin 75   cos 75   sin 15   cos15 
A.
6 3
2
B.
6
2
C.
6
D.
6 3
3
E. 2 6  3
22. In rectangle ABCD, E and F are points on sides AB and DC , respectively, G is the intersection of
segments EC and FB , and H is the intersection of the segments FA and ED . Given that CF=AE=AF=6
and AD=4, find the ratio of the area of EGFH to the area of ABCD.
A. 5 5  12
B.
7 5
50
C.
45  14 5
64
D.
3 5
25
E.
21 5  45
8
23. If S(x) denotes the sum of the positive integral divisors of x, evaluate: S(5)  S(5 2 )  ... S(5 6 )
A. 24,400
B. 24,405
C. 24,411
D. 24,412
E. 24,418
24. Find the number of paths from A to E that only go right and up along the lines of the figure that pass
through the darkened segment closest to A and that do not pass through the darkened segment closest to E.
E
A
A. 188
B. 198
C. 201
D. 207
E. none of the above

25. Evaluate:

k 1
A. 2
n 2
k 2 



 kn  (k  1)  (n  1)  
 (k  1) n 2 (n  1) k 2  

n 1 



B. 3
C. 4
D. 6
E. NOTA
TB1: If x  2 12  3  2 2  4  3 2  ...  17 16 2  18 17 2 , find the product of the digits of x when it is written in
simplified form.
TB2: Rounded to the nearest whole number, what is the expected value of the number of points you score on a
8-question multiple choice test if you answer every question, each question is worth 4 points, and the probability
that you get any given question right is
1
, where n is the question number? (Assume that the probabilities of
n
answering each question correctly are independent.)
TB3: Simplify:
21270544