American Mathematics Competition 12 (AMC 12) — Questions and Answers

Question 1: What is the circumradius of a triangle with sides 3, 4, and 5?

2
3
5/2 (Correct answer)
5

Correct answer: 5/2

R=abc/(4K)=(3*4*5)/(4*6)=60/24=5/2.

Question 2: What is the period of f(x) = sin(3x)?

3pi
2pi
pi/3
2pi/3 (Correct answer)

Correct answer: 2pi/3

The period is 2pi/|b|=2pi/3.

Question 3: How many ways can 3 people be chosen from a group of 8?

336
56 (Correct answer)
28
112

Correct answer: 56

C(8,3)=8!/(3!5!)=56.

Question 4: What is ∑(k=1 to 20) (2k − 1)?

380
400 (Correct answer)
200
420

Correct answer: 400

The sum of the first 20 odd numbers equals 20² = 400.

Question 5: What is the sum of the first 20 positive integers?

190
210 (Correct answer)
200
400

Correct answer: 210

S_20=20*21/2=210.

Question 6: What is the 10th term of the arithmetic sequence 3, 7, 11, 15, ...?

35
43
39 (Correct answer)
47

Correct answer: 39

The common difference is 4, so a₁₀ = 3 + 9 × 4 = 3 + 36 = 39.

Question 7: What is the product of all primitive 6th roots of unity?

0
i
1 (Correct answer)
−1

Correct answer: 1

The primitive 6th roots are e^(iπ/3) and e^(5iπ/3); their product is e^(2πi) = 1.

Question 8: What is the probability of rolling a sum of 7 with two standard dice?

7/36
5/36
1/6 (Correct answer)
1/12

Correct answer: 1/6

The 6 outcomes summing to 7 give P=6/36=1/6.

Question 9: What is the range of arcsin(x)?

[0, π]
[−π, π]
[−π/2, π/2] (Correct answer)
(−π/2, π/2)

Correct answer: [−π/2, π/2]

The arcsin function has range [−π/2, π/2] by convention as the principal value.

Question 10: What is gcd(48, 36)?

6
4
12 (Correct answer)
24

Correct answer: 12

48=36*1+12 and 36=12*3+0, so gcd=12.

Question 11: If a fair die is rolled three times, what is the probability all three rolls are different?

5/9 (Correct answer)
1/6
5/6
1/36

Correct answer: 5/9

P=(6/6)*(5/6)*(4/6)=120/216=5/9.

Question 12: What is lcm(12, 18)?

72
6
216
36 (Correct answer)

Correct answer: 36

lcm(12,18)=12*18/gcd(12,18)=216/6=36.

Question 13: What is the sum of all cube roots of unity?

−1
1
3
0 (Correct answer)

Correct answer: 0

The cube roots of unity 1, ω, ω² are roots of z³ − 1 = 0, and by Vieta's formulas their sum equals 0.

Question 14: How many two-digit prime numbers exist?

15
21 (Correct answer)
25
18

Correct answer: 21

There are 21 primes between 11 and 97: 11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97.

Question 15: How many ways can 3 people be seated in a row of 5 chairs?

20
60 (Correct answer)
120
30

Correct answer: 60

P(5, 3) = 5!/(5−3)! = 5 × 4 × 3 = 60.

Question 16: The roots of x^2 - 5x + 6 = 0 are r and s. What is r^2 + s^2?

13 (Correct answer)
7
25
12

Correct answer: 13

By Vieta's, r+s=5 and rs=6, so r^2+s^2=(r+s)^2-2rs=25-12=13.

Question 17: The polynomial x^3-6x^2+11x-6 has roots r, s, t. Find r+s+t.

3
11
6 (Correct answer)
-6

Correct answer: 6

By Vieta's formulas, the sum of roots equals 6.

Question 18: What is the product (2 + 3i)(1 − 2i)?

−4 + 7i
8 + i
2 − i
8 − i (Correct answer)

Correct answer: 8 − i

Expanding: 2 − 4i + 3i − 6i² = 2 − i + 6 = 8 − i, since i² = −1.

Question 19: The diagonals of a rhombus are 6 and 8. What is its side length?

10
4
7
5 (Correct answer)

Correct answer: 5

The half-diagonals are 3 and 4; side=sqrt(9+16)=5.

Question 20: Using the law of sines in triangle ABC with A = 30°, a = 5, B = 90°, what is b?

5/√3
5√2
5√3
10 (Correct answer)

Correct answer: 10

By the law of sines: 5/sin30° = b/sin90° → 10 = b.

Question 21: What is the probability that a card drawn from a standard 52-card deck is a face card or a heart?

22/52
9/26
11/26 (Correct answer)
25/52

Correct answer: 11/26

By inclusion-exclusion: (12 + 13 − 3)/52 = 22/52 = 11/26.

Question 22: A rectangle has perimeter 34 and area 60. What is the length of its diagonal?

13 (Correct answer)
11
17
15

Correct answer: 13

l+w=17 and lw=60 gives l=12,w=5; diagonal=sqrt(144+25)=13.

Question 23: If z = cos(π/6) + i·sin(π/6), what is z + z̄ (z plus its complex conjugate)?

1
2
√3 (Correct answer)
0

Correct answer: √3

z = √3/2 + i/2 and z̄ = √3/2 − i/2, so z + z̄ = √3 (the imaginary parts cancel).

Question 24: What is the smallest positive integer with exactly 6 divisors?

6
24
18
12 (Correct answer)

Correct answer: 12

12=2^2*3 has (2+1)(1+1)=6 divisors, and no smaller positive integer has exactly 6.

Question 25: For how many integers n with 1 ≤ n ≤ 100 is iⁿ + i^(−n) = 0?

25
75
100
50 (Correct answer)

Correct answer: 50

iⁿ + i^(−n) = 0 exactly when n is odd: for odd n, iⁿ = i or −i and i^(−n) is its opposite, giving 50 values.

American Mathematics Competition 12 (AMC 12)

The AMC 12 is a prestigious 25-question math competition for high school students covering the full pre-calculus curriculum including algebra, geometry, number theory, and combinatorics. A strong score qualifies students for the AIME and potentially the USA Mathematical Olympiad (USAMO).

Questions

75 min

Time Limit

65%

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