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

Question 1: What is i^23?

1
−1
i
−i (Correct answer)

Correct answer: −i

Since i repeats with period 4 and 23 ≡ 3 (mod 4), i^23 = i^3 = −i.

Question 2: What is the remainder when 7^100 is divided by 5?

2
3
1 (Correct answer)
4

Correct answer: 1

7=2(mod5); powers of 2 mod 5 cycle as 2,4,3,1 with period 4; since 100=0(mod4), remainder is 1.

Question 3: A sequence has aₙ = 2n − 1. What is a₁ + a₂ + ... + a₁₀?

100 (Correct answer)
90
55
110

Correct answer: 100

This is the sum of the first 10 odd numbers, which equals 10² = 100.

Question 4: What is the sum of k^2 for k=1 to 5?

25
45
30
55 (Correct answer)

Correct answer: 55

1+4+9+16+25=55.

Question 5: From a 52-card deck, what is the probability of drawing an ace or a king?

1/13
2/13 (Correct answer)
4/52
1/26

Correct answer: 2/13

8 favorable cards (4 aces + 4 kings) out of 52 gives 8/52=2/13.

Question 6: A committee of 3 is chosen from 7 people. How many ways can this be done?

42
35 (Correct answer)
21
28

Correct answer: 35

C(7, 3) = 7!/(3! × 4!) = 35.

Question 7: What is tan(45°)?

√3
1 (Correct answer)
0
1/√3

Correct answer: 1

tan(45°) = sin(45°)/cos(45°) = (√2/2)/(√2/2) = 1.

Question 8: A ball dropped from 10m rebounds to 3/5 of the previous height each bounce. What is the total distance traveled?

35
50
25
40 (Correct answer)

Correct answer: 40

Total=10+2*(6/(1-3/5))=10+30=40 meters.

Question 9: If sin(θ) = 3/5 and θ is in quadrant I, what is cos(θ)?

3/4
5/4
1/5
4/5 (Correct answer)

Correct answer: 4/5

By the Pythagorean identity, cos(θ) = √(1 − 9/25) = √(16/25) = 4/5.

Question 10: Which term of the arithmetic sequence 4, 7, 10, 13, ... is equal to 100?

35th
34th
32nd
33rd (Correct answer)

Correct answer: 33rd

4 + (n−1) × 3 = 100 gives 3(n−1) = 96, so n = 33.

Question 11: If a+b=5 and ab=4, find a^3+b^3.

45
25
80
65 (Correct answer)

Correct answer: 65

a^3+b^3=(a+b)^3-3ab(a+b)=125-60=65.

Question 12: What is 0! (zero factorial)?

0
1 (Correct answer)
−1
Undefined

Correct answer: 1

By convention (and the gamma function), 0! = 1.

Question 13: What is the maximum value of y = 2cos(x) - 1?

3
2
-1
1 (Correct answer)

Correct answer: 1

cos(x) achieves maximum 1, giving y_max=2(1)-1=1.

Question 14: What is the remainder when 2^200 is divided by 3?

0
2
1 (Correct answer)
4

Correct answer: 1

2^(even)=1(mod3) since 2^2=1(mod3); 200 is even so remainder is 1.

Question 15: If z = 1 + i, what is z^4?

−4 (Correct answer)
4i
−4i
4

Correct answer: −4

z² = (1+i)² = 2i, so z⁴ = (2i)² = 4i² = −4.

Question 16: In a right triangle with hypotenuse 10 and one angle 30°, the side opposite that angle has length:

5√3
2
10/√3
5 (Correct answer)

Correct answer: 5

Opposite = hypotenuse × sin(30°) = 10 × 1/2 = 5.

Question 17: What is the amplitude of y = -3sin(2x)?

3 (Correct answer)
-3
2
6

Correct answer: 3

The amplitude is the absolute value of the leading coefficient: |-3|=3.

Question 18: What is the sum of all positive divisors of 28?

56 (Correct answer)
42
14
28

Correct answer: 56

28=2^2*7; sum of divisors=(1+2+4)(1+7)=7*8=56.

Question 19: What is the sum of all cube roots of unity?

−1
0 (Correct answer)
1
3

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 20: How many perfect squares are strictly between 100 and 400?

12
8
10
9 (Correct answer)

Correct answer: 9

From 11^2=121 to 19^2=361, there are 19-11+1=9 perfect squares.

Question 21: What is the sum of the arithmetic series 5 + 8 + 11 + ... + 50?

455
425
490
440 (Correct answer)

Correct answer: 440

There are (50-5)/3+1=16 terms; sum=16*(5+50)/2=8*55=440.

Question 22: A fair die is rolled. What is the probability of getting a number greater than 4?

1/6
1/3 (Correct answer)
2/3
1/2

Correct answer: 1/3

The favorable outcomes are {5, 6}, so probability = 2/6 = 1/3.

Question 23: How many ways can 4 books be chosen from 10 if order does not matter?

210 (Correct answer)
840
120
5040

Correct answer: 210

C(10,4)=10!/(4!6!)=210.

Question 24: If z satisfies z + |z| = 2 + i, what is |z|?

√2
2
5/4 (Correct answer)
3/4

Correct answer: 5/4

Writing z = x + yi gives y = 1 and x + √(x²+1) = 2; solving yields x = 3/4 and |z| = 2 − 3/4 = 5/4.

Question 25: What is the exact value of cos(75 degrees)?

(sqrt(6)-sqrt(2))/4 (Correct answer)
sqrt(3)/2
(sqrt(6)+sqrt(2))/4
sqrt(2)/2

Correct answer: (sqrt(6)-sqrt(2))/4

cos(75)=cos(45+30)=cos45*cos30-sin45*sin30=(sqrt(6)-sqrt(2))/4.

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%

To Pass

Checking for saved progress...

Exam Rules

You can skip questions and return to them later
Flag questions for review before submitting
No feedback shown until you submit the entire exam
Unanswered questions count as wrong — answer everything
10 pretest questions are mixed in and don't affect your score
Timer auto-submits when time runs out
Your progress is auto-saved every 30 seconds