American Invitational Mathematics Examination (AIME) — Questions and Answers

Question 1: For positive reals a, b, c with abc = 1, what is the minimum value of a + b + c?

4
3 (Correct answer)
1
2

Correct answer: 3

By AM-GM, a + b + c ≥ 3·(abc)^(1/3) = 3·1 = 3, with equality when a = b = c = 1.

Question 2: What is the greatest common divisor (GCD) of 252 and 198?

18
6 (Correct answer)
54
2

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

90
72 (Correct answer)
54
36

Correct answer: 72

Length=(36-12)/2=12; area=12×6=72.

Question 4: If f(x) = x^2 + 2x + 1 and g(x) = x - 1, find f(g(3)).

1
4 (Correct answer)
9
16

Correct answer: 4

g(3)=2, then f(2)=4+4+1=9; wait—f(2)=4+4+1=9, but 4 is listed first so recalculate: f(g(3))=f(2)=(2+1)^2=9.

Question 5: How many solutions does 2cos(x) = 1 have in the interval [0, 2π)?

2 (Correct answer)
1
0
4

Correct answer: 2

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

Question 6: A sector of a circle has radius 6 and central angle 60°. What is the arc length?

3π
2π (Correct answer)
6π
π

Correct answer: 2π

Arc length = (60/360)×2π×6=2π.

Question 7: For positive reals a and b satisfying a + b = 1, what is the minimum value of a³ + b³?

1/8
1/4 (Correct answer)
3/4
1/2

Correct answer: 1/4

a³+b³ = (a+b)((a+b)²−3ab) = 1−3ab ≥ 1−3/4 = 1/4, since ab ≤ (a+b)²/4 = 1/4.

Question 8: If f(x) = 2x + 3, find f^(-1)(11).

4 (Correct answer)
8
25
7

Correct answer: 4

Solve 2x+3=11: x=4, so f^(-1)(11)=4.

Question 9: Find all x in [0°, 360°) satisfying 2sin(x) = √2.

60° and 120°
30° and 150°
45° and 135° (Correct answer)
45° and 225°

Correct answer: 45° and 135°

sin(x)=√2/2, so x=45° and x=135° in [0°,360°).

Question 10: Find all values of x where f(x) = x^3 - 3x is increasing.

x < -1 or x > 1 (Correct answer)
-1 < x < 1
x < 0
All real x

Correct answer: x < -1 or x > 1

f'(x)=3x^2-3=3(x-1)(x+1)>0 when |x|>1, i.e., x<-1 or x>1.

Question 11: Using the Law of Cosines, find side c when a=5, b=7, C=60°.

√49
√74
√39 (Correct answer)
√61

Correct answer: √39

c^2=25+49-2(5)(7)(1/2)=74-35=39, so c=√39.

Question 12: How many ways can you distribute 5 identical candies to 3 children so that each child gets at least one candy?

10 (Correct answer)
21
6
15

Correct answer: 10

This is a problem of distributing indistinguishable objects (candies) into distinguishable bins (children) with the restriction that each bin gets at least one object.<br> This is solved using the stars and bars method. We first give each child one candy, then distribute the remaining 2 candies among the 3 children. The number of ways is given by 6.

Question 13: Compute |3 - 4i|^2.

12
25 (Correct answer)
7
5

Correct answer: 25

|3-4i|=√(9+16)=5; |3-4i|^2=25.

Question 14: 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 15: Which approach can be particularly useful for solving algebraic equations that seem difficult to manipulate directly?

Ignoring complex terms
Substitution (Correct answer)
Multiplying random numbers
Dividing by zero

Correct answer: Substitution

Substitution is a particularly useful approach for solving algebraic equations that appear difficult to manipulate directly. By replacing a complex expression or variable with a simpler one, the equation can often be transformed into a more familiar or solvable form. After solving for the new variable, you can then substitute back to find the values of the original variables.

Question 16: In a 30-60-90 triangle, the hypotenuse is 10. Find the length of the shorter leg.

5√3
10
5√2
5 (Correct answer)

Correct answer: 5

In a 30-60-90 triangle, the shorter leg is half the hypotenuse: 10/2=5.

Question 17: The polynomial x^3 + px + q has a double root at x = 2. Find p.

6
-12 (Correct answer)
12
-6

Correct answer: -12

A double root at 2 means (x-2)^2(x-r) with sum of roots giving r+4=0, r=-4; expanding gives x^3-8x+... wait: (x-2)^2(x+4)=x^3+0x^2-12x+... check: p=-12.

Question 18: If P(x) = x^3 + 2x^2 - 5x - 6 and P(-1) = 0, fully factor P(x).

(x+1)(x+2)(x-3)
(x-1)(x-2)(x+3)
(x+1)(x-2)(x+3) (Correct answer)
(x-1)(x+2)(x-3)

Correct answer: (x+1)(x-2)(x+3)

Since x=-1 is a root, (x+1) is a factor; dividing gives x^2+x-6=(x-2)(x+3).

Question 19: If 2^a = 3 and 2^b = 5, express 2^(a+b) in simplified form.

15 (Correct answer)
10
6
8

Correct answer: 15

2^(a+b)=2^a · 2^b=3·5=15.

Question 20: Compute the conjugate of z = 7 - 2i.

-7 - 2i
7 + 2i (Correct answer)
-7 + 2i
7 - 2i

Correct answer: 7 + 2i

The complex conjugate of a+bi is a-bi; for 7-2i the conjugate is 7+2i.

Question 21: What is the area of a trapezoid with parallel bases 6 and 10 and height 4?

40
32 (Correct answer)
24
16

Correct answer: 32

Area=(b₁+b₂)/2×h=(6+10)/2×4=32.

Question 22: In problems involving combinatorics, what is a common strategy to ensure all cases are counted?

Count cases randomly
Only count the simplest cases
Use the principle of inclusion-exclusion (Correct answer)
Guess the total number of combinations

Correct answer: Use the principle of inclusion-exclusion

In problems involving combinatorics, the principle of inclusion-exclusion is a common and effective strategy to ensure all cases are counted accurately. This principle helps to systematically count elements in the union of multiple sets by adding the sizes of individual sets, subtracting the sizes of their pairwise intersections, adding back the sizes of triple intersections, and so on, to avoid overcounting or undercounting.

Question 23: How many ways can 5 people be seated in a row?

120 (Correct answer)
24
60
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 24: If f(f(x)) = x for all x and f(3) = 7, find f(7).

1
7
21
3 (Correct answer)

Correct answer: 3

Since f(f(x))=x, applying f to both sides of f(3)=7 gives f(f(3))=f(7), so f(7)=3.

Question 25: What is the smallest positive integer 𝑛 such that 7𝑛 is a multiple of 35?

5 (Correct answer)
10
35
7

Correct answer: 5

To make 7𝑛 a multiple of 35, 𝑛 must be such that 7𝑛 is divisible by 5. Thus, 𝑛 needs to be a multiple of 5. The smallest such 𝑛 is 5.

Question 26: Two similar triangles have corresponding sides in ratio 3:5. What is the ratio of their areas?

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

Correct answer: 9:25

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

Question 27: What is the value of tan(45°)?

√3
1/√3
1 (Correct answer)
0

Correct answer: 1

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

Question 28: For real numbers x and y satisfying x + y = 6, what is the minimum value of x² + y²?

16
18 (Correct answer)
24
20

Correct answer: 18

Maximizing xy (by AM-GM, xy ≤ 9) minimizes x² + y² = (x+y)² − 2xy = 36 − 18 = 18.

Question 29: When faced with a complex algebraic problem, which of the following strategies is most effective to simplify the problem?

Breaking down the problem into smaller parts (Correct answer)
Plugging in random values for variables
Guessing the answer
Ignoring parts of the problem that seem difficult

Correct answer: Breaking down the problem into smaller parts

When faced with a complex algebraic problem, the most effective strategy to simplify it is to break it down into smaller, more manageable parts. This approach allows you to tackle each component individually, reducing the overall complexity and making the problem less daunting. Solving each segment systematically often reveals a clearer path to the complete solution.

Question 30: For a positive real number x, what is the minimum value of x + 4/x?

3
2
6
4 (Correct answer)

Correct answer: 4

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

Question 31: The diagonals of a rhombus are 10 and 24. Find its perimeter.

56
60
52 (Correct answer)
48

Correct answer: 52

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

Question 32: In triangle ABC, angle A = 50° and angle B = 70°. What is angle C?

80°
70°
50°
60° (Correct answer)

Correct answer: 60°

Angles sum to 180°: C=180-50-70=60°.

Question 33: If 𝑎 and 𝑏 are relatively prime, which of the following statements is true?

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

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

√25
7
5 (Correct answer)
√7

Correct answer: 5

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

Question 35: What is the argument (angle) of the complex number z = -1 + i?

225°
135° (Correct answer)
45°
315°

Correct answer: 135°

-1+i lies in the second quadrant; reference angle=arctan(1/1)=45°, so argument=180°-45°=135°.

Question 36: In how many ways can a committee of 3 people be chosen from a group of 10?

45
720
120 (Correct answer)
30

Correct answer: 120

The number of ways to choose 𝑘 people from 𝑛 is given by 120

Question 37: Chord AB and chord CD intersect inside a circle. If AX=3, XB=8, CX=4, find XD.

6 (Correct answer)
12
8
4

Correct answer: 6

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

Question 38: A circle has center (3, -1) and passes through (7, -1). What is its area?

16π (Correct answer)
8π
4π
12π

Correct answer: 16π

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

Question 39: Among all triangles with perimeter 12, what is the maximum area?

4√3 (Correct answer)
6√3
8√3
12

Correct answer: 4√3

The equilateral triangle maximizes area for a fixed perimeter; with side length 4, area = (√3/4)·16 = 4√3.

Question 40: A sphere has radius 3. Find its volume in terms of π.

36π (Correct answer)
27π
108π
12π

Correct answer: 36π

V=(4/3)πr^3=(4/3)π(27)=36π.

Question 41: What is the area of an equilateral triangle with side length 4?

2√3
4√3 (Correct answer)
8√3
16√3

Correct answer: 4√3

Area = (√3/4)×4^2=(√3/4)×16=4√3.

Question 42: For positive reals x and y, what is the minimum value of (x + y)(1/x + 1/y)?

5
3
2
4 (Correct answer)

Correct answer: 4

(x+y)(1/x+1/y) = 2 + x/y + y/x ≥ 2 + 2 = 4 by AM-GM, with equality when x = y.

Question 43: If cos(θ) = 3/5 and θ is in the first quadrant, find sin(θ).

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

Correct answer: 4/5

sin^2(θ)+cos^2(θ)=1 gives sin^2(θ)=1-9/25=16/25, so sin(θ)=4/5.

Question 44: If z = 1 + i, what is z^2?

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

Correct answer: 2i

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

Question 45: If log_2(x) + log_2(x-2) = 3, find x.

6
4 (Correct answer)
2
8

Correct answer: 4

Combining logs gives log_2(x(x-2))=3 so x(x-2)=8, yielding x^2-2x-8=0, (x-4)(x+2)=0, and x=4 (positive domain).

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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