Mock AMC 2007
1. At West High School, students need at least 22 credits to graduate. Freshmen and sophomores must have either 6 or 7 credits, and juniors and seniors must have 5 to 7 credits. How
many ways can a student successfully graduate in 4 years with 22 or more credits?
A. 4 B. 10 C. 12 D. 24 E. 36
2. Dirk Nowitzki plays basketball for the Dallas Mavericks, but he is from Germany. The metric
system is used in Germany. He is asked how high a basketball is, and replies, ”About 3
meters.” How high is 3 meters in feet?(inch=2.54cm)
A. 9 ft 10 in B. 9 ft 11 in C. 10 ft D. 10 ft 1 in E. 10 ft 2 in
3. Mustafa has 830 songs on his iPod. He wants 1000 songs on his iPod. By what percent must
he increase his present total number of songs to get more than 1000 songs?
A. 199% B. 20% C. 120% D. 21% E. 121%
4. Biff and Beagle have lots of skateboards, but Beagle has more boards than Biff. Let B be the
number of boards Beagle has, and b be the number of boards Biff has.
Bb = 156 and B 2 + b2 = 313.
Assuming that all boards are distinguishable, in how many ways can you choose 5 of Beagle’s
boards and 3 of Biff’s Boards?
A. 203860800 B. 163088640 C. 183100 D. 237600 E. 283140
5. The Euler’s totient function φ(x) is defined to be the number of positive integers less than or
equal to n and relatively prime to n. For example, φ(16) = 8, because 1,3,5,7,9,11,13, and 15
are relatively prime to 16. For how many natural numbers is φ(x) < 5?
A. 7 B. 9 C. 11 D. 13 E. 15
6. What is the probability that the square of a randomly picked integer will give a remainder of
2,4, or 6 when divided by 7?
A. 21 B. 37 C. 0 D. 47 E. 27
7. What is the largest integer less than 100000 for which
A. 64 B. 4096 C. 15625 D. 46656 E. 65536
√
2
x and
√
3
x are both integers?
8. In Wisconsin, it is cold and snowy. If the temperature goes below 32◦ Fahrenheit, there is a
35% chance of snow. The number of times it goes below 32◦ is equal to the product of the
first four odd numbers. What is the expected number of days that it will snow?
A. 3 B. 33 C. 35 D. 37 E. 45
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Mock AMC 2007
9. What is the average of the largest, smallest, and median of the fractions
2006
13771 ?
47849902
25924951
15
47849902
B. 25924951
C. 329181984
D. 329181984
E. 103
A. 109727328
5486300
12 14 15 192
83 , 96 , 103 , 1318
and
10. Define function " such that x " y = xy + xy for x #= 0, y #= 0.
What is (4 " 3) " (6 " 5) + (2 " 1) " (−2 " −3)
89392056
B. 213885220
C. 2393891
D. 4 E. 634842
A. 11969455
2973750
594759
184381
11. What is the surface area of a cube inscribed in a sphere inscribed in a cube with surface area
54? √
√
√
A. 6 3 B. 12 C. 272 3 D. 18 E. 12 3
12. How many ways can Beagle skateboard on the lines starting at point A, high five Biff at point
B, and get to point C going only down or to the right?
A. 350
B. 343
C. 333
D. 330
E. 324
13. An equilateral triangle is inscribed in a circle, which is inscribed in a square of side length
100. What is the area
√ of the triangle?
√
√
√
√
5000 3
5000 3
C.
D. 2500 3 E. 1225 3
A. 1875 3 B.
3
2
14. What is log6 1 + log10 2 + log12 3 + log12 4 + log10 5 + log6 6 equal to?
A. 2 + log10 12 B. 2 C. 3 D. 4 E. 1 + log10 21
15. Find
A.
!
√15
43
5+
1
4+
B.
3+
1
1
2+ 1
1
√
15 43
43
√
C. 43 15
D.
√
645
16. What are the last 4 digits of 8945 ?
A. 9191 B. 3051 C. 9449 D. 5049
17. What is 1839B14 in base 23?
A. 50B523 B. 6109323 C. 5B523
E. 645
E. 4989
D. 500B523
2
E. CAB23
Mock AMC 2007
18. There is a system for pricing this menu.
Soup $18
Salad $23
Roast Beef $41
Ice Cream $36
Cof f ee $27
How much would oysters cost? (y is not a vowel)
A. $30 B. $33 C. $36 D. $39 E. $42
19. If ∠CAB = 50◦ ,∠DCB = 90◦ , arcAB = arcCD, then find ∠CEB.
A. 80◦
B. 95◦
C. 100◦
D. 110◦
E. 140◦
20. What is the volume of a regular octahedron with side length 15?
√
A.75 3
√
B. 75 3/2
√
C. 1125 3/2
√
D. 1125 3
√
E. 150 3
21. Given that ∠ABC = ∠CDE = ∠DEG = ∠EF A = ∠F GA = 90◦ , AC = 18000 and
CB = 10800. Find the length of F G.
A. 5529.6
B. 4423.68
Π5 k 2
22. Find "k=2
5
2
k=2 k
11
2880
B. 2880
A. 11
C.
C. 3538.944
800
3
D.
3
800
D. 6912
E.
18
5
3
E. 7212.86
Mock AMC 2007
23. Let the uberset of a number be the set of all numbers that can be made by rearranging the
digits of a number. For example, the uberset of the number 380 is 380, 308, 83, 38, 830, 803.
What is the third number greater than 100 for which all of the elements of that number’s
uberset are prime?
A. 131 B. 133 C. 137 D. 191 E. 311
#
#
√
24. If 9x − 7 =
3y + 2 and 3 y −
2(x + 1) + 3 3y − 2(4x − 7) = 3 10, what is x?
√
√
A.3 B. 3 10 C. 4 D. 3 30 E. 5
25. On the ”metric clock,” there are 100 seconds in a minute, 100 minutes in an hour, and 10
hours in a day. Right now, it is 7:85. What is the next time the hour and minute hand make
a 155◦ ?
A. 7:65:40 B. 8:43:69 C. 8:25:62 D. 8:24:69 E. 8:42:38
4