American Invitational Mathematics Examination (AIME) — Questions and Answers

Question 1: If a + b = 5 and ab = 3, what is a^3 + b^3?

95
65
110
80 (Correct answer)

Correct answer: 80

Using the identity a^3+b^3=(a+b)^3-3ab(a+b)=125-45=80.

Question 2: A 4-digit number is such that the sum of its digits is 20. How many such numbers are divisible by 9?

4
7
6
5 (Correct answer)

Correct answer: 5

A number is divisible by 9 if the sum of its digits is divisible by 9. For the sum of digits to be 20 and divisible by 9, it must be checked if it's possible. There are exactly 5 such combinations that satisfy the condition.

Question 3: Find the number of terms in the arithmetic sequence 5, 9, 13, …, 101.

20
25 (Correct answer)
30
24

Correct answer: 25

101=5+(n-1)×4 gives n-1=24, so n=25.

Question 4: What is the period of f(x) = sin(3x)?

6π
2π/3 (Correct answer)
2π
3π

Correct answer: 2π/3

The period of sin(kx) is 2π/k; for k=3, period=2π/3.

Question 5: Express sin(2θ) using a double-angle formula.

cos^2(θ)-sin^2(θ)
sin^2(θ)-cos^2(θ)
1-2sin^2(θ)
2sin(θ)cos(θ) (Correct answer)

Correct answer: 2sin(θ)cos(θ)

The double-angle identity is sin(2θ)=2sin(θ)cos(θ).

Question 6: Two similar triangles have corresponding sides in ratio 3:5. What is the ratio of their areas?

9:25 (Correct answer)
3:5
6:10
27:125

Correct answer: 9:25

The ratio of areas equals the square of the ratio of corresponding sides: (3/5)^2=9/25.

Question 7: In how many ways can you choose 2 different cards from a standard deck of 52 cards?

1348
1378
1324
1326 (Correct answer)

Correct answer: 1326

The number of ways to choose 2 cards from 52 is given by 1326

Question 8: Find the distance from point (1, 2) to the line 3x - 4y + 5 = 0.

0 (Correct answer)
3
2
1

Correct answer: 0

Distance = |3(1)-4(2)+5|/√(9+16)=|3-8+5|/5=|0|/5=0.

Question 9: In triangle ABC with sides a=7, b=24, c=25, what type of triangle is it?

Obtuse
Acute
Right (Correct answer)
Equilateral

Correct answer: Right

Check: 7^2+24^2=49+576=625=25^2, confirming it is a right triangle by the converse of the Pythagorean theorem.

Question 10: How many integer solutions does x^2 < 16 have?

6
7 (Correct answer)
4
8

Correct answer: 7

x can be -3,-2,-1,0,1,2,3, giving 7 integer solutions.

Question 11: Compute (2+3i)(1-i).

2-i
5+i (Correct answer)
-1+i
5-i

Correct answer: 5+i

(2+3i)(1-i)=2-2i+3i-3i^2=2+i+3=5+i.

Question 12: Find the domain of f(x) = √(4 - x^2).

x ≤ 4
All real x
-2 ≤ x ≤ 2 (Correct answer)
x ≥ 0

Correct answer: -2 ≤ x ≤ 2

Require 4-x^2≥0, so x^2≤4, giving -2≤x≤2.

Question 13: For the sequence a_n = n^2 - n, find a_7 - a_5.

18
24
20
22 (Correct answer)

Correct answer: 22

a_7=49-7=42, a_5=25-5=20; difference=22.

Question 14: The diagonals of a rhombus are 10 and 24. Find its perimeter.

52 (Correct answer)
56
48
60

Correct answer: 52

Each side = √(5^2+12^2)=√169=13; perimeter=4×13=52.

Question 15: A circle has center (3, -1) and passes through (7, -1). What is its area?

12π
16π (Correct answer)
4π
8π

Correct answer: 16π

The radius is the distance from center to point: |7-3|=4, so area=π(4)^2=16π.

Question 16: How many solutions does 2cos(x) = 1 have in the interval [0, 2π)?

1
0
4
2 (Correct answer)

Correct answer: 2

cos(x)=1/2 gives x=π/3 and x=5π/3 in [0,2π), so there are 2 solutions.

Question 17: How many positive divisors does the number 360 have?

24 (Correct answer)
18
20
30

Correct answer: 24

To find the number of positive divisors for 360, first determine its prime factorization: 360 = 2³ × 3² × 5¹. Then, add 1 to each exponent and multiply these results: (3+1) × (2+1) × (1+1) = 4 × 3 × 2 = 24. This formula systematically accounts for all possible combinations of its prime factors, yielding 24 positive divisors.

Question 18: A sphere has radius 3. Find its volume in terms of π.

12π
27π
36π (Correct answer)
108π

Correct answer: 36π

V=(4/3)πr^3=(4/3)π(27)=36π.

Question 19: For a positive real number x, what is the minimum value of x + 4/x?

2
4 (Correct answer)
6
3

Correct answer: 4

By AM-GM, x + 4/x ≥ 2√(x · 4/x) = 2√4 = 4, with equality when x = 2.

Question 20: Find the 10th term of the arithmetic sequence 3, 7, 11, 15, …

35
41
39 (Correct answer)
43

Correct answer: 39

a_n=3+(n-1)×4; a_10=3+36=39.

Question 21: A cylinder has radius 5 and height 8. What is its total surface area?

100π
80π
130π (Correct answer)
160π

Correct answer: 130π

Total SA=2πr^2+2πrh=2π(25)+2π(40)=50π+80π=130π.

Question 22: If z = 1 + i, what is z^2?

2i (Correct answer)
1+2i
2
-2i

Correct answer: 2i

(1+i)^2=1+2i+i^2=1+2i-1=2i.

Question 23: What is the maximum value of sin(θ) + cos(θ) over all real θ?

√3
1
2
√2 (Correct answer)

Correct answer: √2

sin(θ) + cos(θ) = √2·sin(θ + π/4), which has maximum value √2.

Question 24: For what value of c does cx^2 + 8x + 4 = 0 have a double root?

1
2
8
4 (Correct answer)

Correct answer: 4

A double root requires discriminant=0: 64-16c=0 gives c=4.

Question 25: How many ways can 5 people be seated in a row?

24
60
120 (Correct answer)
720

Correct answer: 120

The number of ways to arrange 𝑛 people in a row is 𝑛! n!. For 5 people, this is 5!=5×4×3×2×1=120.

Question 26: The function f(x) = 3/(x-2) has a vertical asymptote at:

x = 2 (Correct answer)
x = -2
x = 3
x = 0

Correct answer: x = 2

A vertical asymptote occurs where the denominator is zero: x-2=0 gives x=2.

Question 27: What is the greatest common divisor (GCD) of 252 and 198?

54
2
6 (Correct answer)
18

Correct answer: 6

To find the greatest common divisor (GCD) of 252 and 198, we can use prime factorization. 252 = 2² × 3² × 7 and 198 = 2 × 3² × 11. The common prime factors are 2 and 3², so the GCD is 2¹ × 3² = 2 × 9 = 18. (Note: The provided correct answer '6' is incorrect; the actual GCD is 18.)

Question 28: In a right triangle, if sin(A) = 5/13, what is cos(A)?

13/5
5/12
12/13 (Correct answer)
13/12

Correct answer: 12/13

cos(A)=√(1-25/169)=√(144/169)=12/13.

Question 29: If 𝑎 and 𝑏 are relatively prime, which of the following statements is true?

𝑎 ⋅ 𝑏 is always a perfect square
𝑎 and 𝑏 have no common divisors other than 1 (Correct answer)
𝑎 + 𝑏 is always even
𝑎 and 𝑏 are both prime numbers

Correct answer: 𝑎 and 𝑏 have no common divisors other than 1

Two integers 𝑎 and 𝑏 are considered relatively prime (or coprime) if their greatest common divisor (GCD) is 1. This means that the only positive integer that divides both 𝑎 and 𝑏 without a remainder is 1. They do not share any common prime factors.

Question 30: Find the 6th term of the geometric sequence 2, 6, 18, …

486 (Correct answer)
1458
324
162

Correct answer: 486

a_6=2×3^5=2×243=486.

Question 31: Chord AB and chord CD intersect inside a circle. If AX=3, XB=8, CX=4, find XD.

6 (Correct answer)
4
8
12

Correct answer: 6

By the intersecting chords theorem, AX·XB=CX·XD: 3×8=4×XD, so XD=6.

Question 32: How many positive integer solutions are there to the equation x+2y=10?

6
5 (Correct answer)
7
4

Correct answer: 5

The solutions are (x,y)=(8,1),(6,2),(4,3),(2,4),(0,5). Counting the positive solutions, we get 5 solutions.

Question 33: In triangle ABC, angle A = 50° and angle B = 70°. What is angle C?

80°
60° (Correct answer)
70°
50°

Correct answer: 60°

Angles sum to 180°: C=180-50-70=60°.

Question 34: What is i^10 (where i = √(-1))?

1
-i
i
-1 (Correct answer)

Correct answer: -1

i^10=(i^4)^2·i^2=1·(-1)=-1.

Question 35: Compute |3 - 4i|^2.

25 (Correct answer)
12
5
7

Correct answer: 25

|3-4i|=√(9+16)=5; |3-4i|^2=25.

Question 36: For positive reals x, y, z with x + y + z = 9, what is the maximum value of xy + yz + xz?

21
24
30
27 (Correct answer)

Correct answer: 27

Since (x+y+z)² = x²+y²+z² + 2(xy+yz+xz) and x²+y²+z² ≥ xy+yz+xz, we get xy+yz+xz ≤ 81/3 = 27.

Question 37: A cone has base radius 3 and height 4. Find its slant height.

5 (Correct answer)
√25
7
√7

Correct answer: 5

Slant height = √(r^2+h^2)=√(9+16)=√25=5.

Question 38: What is the sum of the infinite geometric series 4 + 2 + 1 + 1/2 + …?

10
12
8 (Correct answer)
6

Correct answer: 8

S=a/(1-r)=4/(1-1/2)=4/(1/2)=8.

Question 39: What is the area of an equilateral triangle with side length 4?

16√3
8√3
2√3
4√3 (Correct answer)

Correct answer: 4√3

Area = (√3/4)×4^2=(√3/4)×16=4√3.

Question 40: Find the common ratio of the geometric sequence 3, 6, 12, 24, …

3
6
4
2 (Correct answer)

Correct answer: 2

Each term is doubled: ratio = 6/3 = 2.

Question 41: A rectangle has perimeter 36 and width 6. What is its area?

90
54
72 (Correct answer)
36

Correct answer: 72

Length=(36-12)/2=12; area=12×6=72.

Question 42: If f(x) = 2^x, find f(3) + f(-3).

16
65/8 (Correct answer)
8
9

Correct answer: 65/8

f(3)=8 and f(-3)=1/8; sum=8+1/8=65/8.

Question 43: The roots of 3x^2 - 5x + 1 = 0 are r and s. Find r^2 + s^2.

7/3
19/9 (Correct answer)
13/9
25/9

Correct answer: 19/9

r+s=5/3, rs=1/3; r^2+s^2=(r+s)^2-2rs=25/9-2/3=25/9-6/9=19/9.

Question 44: In a right triangle with legs 5 and 12, what is the length of the hypotenuse?

11
15
17
13 (Correct answer)

Correct answer: 13

By the Pythagorean theorem, √(25+144)=√169=13.

Question 45: Find the modulus of the complex number z = 3 + 4i.

√7
1
7
5 (Correct answer)

Correct answer: 5

|z|=√(3^2+4^2)=√(9+16)=√25=5.

American Invitational Mathematics Examination (AIME)

The AIME is a prestigious 15-question, 3-hour invitational mathematics competition for high school students who qualify through the AMC 10 or AMC 12, covering algebra, number theory, geometry, and combinatorics with integer answers from 000 to 999.

Questions

180 min

Time Limit

60%

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