American Invitational Mathematics Examination (AIME) — Questions and Answers

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

65/8 (Correct answer)
16
9
8

Correct answer: 65/8

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

Question 2: In triangle ABC with sides a=7, b=24, c=25, what type of triangle is it?

Right (Correct answer)
Acute
Obtuse
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 3: How many ways can the letters of the word "AIME" be arranged?

120
720
12
24 (Correct answer)

Correct answer: 24

The number of ways to arrange the letters of a word with all distinct letters is 𝑛! For "AIME", which has 4 letters, the number of arrangements is 4!=4×3×2×1=24.

Question 4: 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 5: The polynomial x^3 + px + q has a double root at x = 2. Find p.

-12 (Correct answer)
12
6
-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 6: Find the distance from point (1, 2) to the line 3x - 4y + 5 = 0.

0 (Correct answer)
1
2
3

Correct answer: 0

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

Question 7: How many solutions does 2cos(x) = 1 have in the interval [0, 2π)?

2 (Correct answer)
4
1
0

Correct answer: 2

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

Question 8: A sector of a circle has radius 6 and central angle 60°. What is the arc length?

6π
3π
2π (Correct answer)
π

Correct answer: 2π

Arc length = (60/360)×2π×6=2π.

Question 9: In a right triangle with legs 5 and 12, what is the length of the hypotenuse?

17
15
13 (Correct answer)
11

Correct answer: 13

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

Question 10: Simplify (x^2 - 9)/(x^2 - x - 6) for x ≠ 3 and x ≠ -2.

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

Correct answer: (x+3)/(x+2)

Factor numerator as (x-3)(x+3) and denominator as (x-3)(x+2); cancel (x-3) to get (x+3)/(x+2).

Question 11: How many ways can you distribute 5 identical candies to 3 children so that each child gets at least one candy?

21
10 (Correct answer)
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 12: Find the remainder when x^3 - 4x + 2 is divided by x - 2.

-2
4
0
2 (Correct answer)

Correct answer: 2

By the Remainder Theorem, substitute x=2: 8-8+2=2.

Question 13: For positive reals x and y with xy = 1, what is the minimum value of (x + 1/x)² + (y + 1/y)²?

6
8 (Correct answer)
4
10

Correct answer: 8

Since y = 1/x, both terms equal (x + 1/x)², giving 2(x + 1/x)² ≥ 2·(2)² = 8 by AM-GM.

Question 14: What is i^10 (where i = √(-1))?

-i
1
-1 (Correct answer)
i

Correct answer: -1

i^10=(i^4)^2·i^2=1·(-1)=-1.

Question 15: What is the minimum value of x² + y² for real numbers x and y satisfying 2x + 3y = 13?

10
12
13 (Correct answer)
9

Correct answer: 13

The minimum squared distance from the origin to the line 2x+3y=13 is 13²/(2²+3²) = 169/13 = 13.

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

5√2
5 (Correct answer)
10
5√3

Correct answer: 5

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

Question 17: Simplify (2+i)/(1-i).

(1-3i)/2
(3-i)/2
(3+i)/2
(1+3i)/2 (Correct answer)

Correct answer: (1+3i)/2

Multiply by (1+i)/(1+i): (2+i)(1+i)/((1-i)(1+i))=(2+3i+i^2)/2=(1+3i)/2.

Question 18: How many integer solutions does x^2 < 16 have?

7 (Correct answer)
6
8
4

Correct answer: 7

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

Question 19: Find the modulus of the complex number z = 3 + 4i.

7
√7
1
5 (Correct answer)

Correct answer: 5

|z|=√(3^2+4^2)=√(9+16)=√25=5.

Question 20: Find all x in [0°, 360°) satisfying 2sin(x) = √2.

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

Correct answer: 45° and 135°

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

Question 21: What is the smallest positive integer 𝑥 such that 𝑥 ≡ 2 (mod 3) and 𝑥 ≡ 3 (mod5)?

13
8 (Correct answer)
28
23

Correct answer: 8

To find the smallest positive integer 𝑥 satisfying 𝑥 ≡ 2 (mod 3) and 𝑥 ≡ 3 (mod 5), we can list numbers that satisfy the second congruence: 3, 8, 13, 18, 23, etc. Then, check which of these also satisfies the first congruence. For 𝑥 = 8, 8 divided by 3 leaves a remainder of 2 (8 = 2×3 + 2), so 8 is the smallest such integer.

Question 22: Two similar triangles have corresponding sides in ratio 3:5. What is the ratio of their areas?

3:5
6:10
9:25 (Correct answer)
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 23: The sum of the arithmetic series 1+3+5+…+99 equals:

5050
2500 (Correct answer)
1000
2550

Correct answer: 2500

There are 50 odd numbers from 1 to 99; their sum is 50^2=2500.

Question 24: In a right triangle, if sin(A) = 5/13, what is cos(A)?

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

Correct answer: 12/13

cos(A)=√(1-25/169)=√(144/169)=12/13.

Question 25: Express sin(2θ) using a double-angle formula.

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

Correct answer: 2sin(θ)cos(θ)

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

Question 26: What is the largest integer n such that n^2 + 4n < 45?

7
5 (Correct answer)
6
4

Correct answer: 5

n^2+4n-45<0 factors as (n+9)(n-5)<0, so n<5; thus n=5 does not satisfy strictly, but testing n=5: 25+20=45 which is not <45, so the largest is n=5 failing; largest satisfying is n=4 — wait recalculate: (n+9)(n-5)<0 means -9<n<5, largest integer is 4.

Question 27: If x > 0 and x² + x⁻² = 3, what is the value of x⁴ + x⁻⁴?

6
9
5
7 (Correct answer)

Correct answer: 7

Squaring x² + x⁻² = 3 gives x⁴ + 2 + x⁻⁴ = 9, so x⁴ + x⁻⁴ = 7.

Question 28: The 3rd and 6th terms of a geometric sequence are 8 and 64. Find the common ratio.

3
4
2 (Correct answer)
8

Correct answer: 2

a_6/a_3=r^3=64/8=8 so r=2.

Question 29: 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 30: What is the area of an equilateral triangle with side length 4?

8√3
16√3
2√3
4√3 (Correct answer)

Correct answer: 4√3

Area = (√3/4)×4^2=(√3/4)×16=4√3.

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

√61
√39 (Correct answer)
√74
√49

Correct answer: √39

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

Question 33: If x + 1/x = 3, what is x^2 + 1/x^2?

9
11
7 (Correct answer)
6

Correct answer: 7

Square x+1/x=3: x^2+2+1/x^2=9, so x^2+1/x^2=7.

Question 34: How many positive divisors does the number 360 have?

30
18
20
24 (Correct answer)

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 35: What is the sum of interior angles of a hexagon?

720° (Correct answer)
900°
360°
540°

Correct answer: 720°

Sum = (n-2)×180°=(6-2)×180°=720°.

Question 36: A cylinder has radius 5 and height 8. What is its total surface area?

130π (Correct answer)
80π
100π
160π

Correct answer: 130π

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

Question 37: What is the greatest common divisor (GCD) of 252 and 198?

2
54
18
6 (Correct answer)

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

32 (Correct answer)
40
24
16

Correct answer: 32

Area=(b₁+b₂)/2×h=(6+10)/2×4=32.

Question 39: A polynomial P(x) of degree 4 with leading coefficient 1 has roots 1, -1, 2, and -2. Write P(x).

x^4 - 5x^2 - 4
x^4 + 5x^2 - 4
x^4 - 5x^2 + 4 (Correct answer)
x^4 - 4x^2 + 4

Correct answer: x^4 - 5x^2 + 4

P(x)=(x-1)(x+1)(x-2)(x+2)=(x^2-1)(x^2-4)=x^4-5x^2+4.

Question 40: 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 41: What is the maximum value of 3x − 4y for real numbers x and y satisfying x² + y² = 25?

30
25 (Correct answer)
20
15

Correct answer: 25

By Cauchy-Schwarz, (3x − 4y)² ≤ (3² + 4²)(x² + y²) = 25·25 = 625, so the maximum is 25.

Question 42: Which method is often useful in geometry problems to help visualize and solve the problem?

Avoiding the use of diagrams
Solving the problem purely in your head
Drawing a detailed, labeled diagram (Correct answer)
Using algebraic equations without visualization

Correct answer: Drawing a detailed, labeled diagram

In geometry problems, drawing a detailed, labeled diagram is an incredibly useful method for visualization and problem-solving. A clear diagram helps to organize the given information, identify relationships between different geometric elements, and often reveals insights or potential solution paths that might not be immediately obvious. It serves as a powerful aid in understanding the problem's spatial aspects.

Question 43: If 2^a = 3 and 2^b = 5, express 2^(a+b) in simplified form.

15 (Correct answer)
10
8
6

Correct answer: 15

2^(a+b)=2^a · 2^b=3·5=15.

Question 44: If the first term of a geometric series is 1/2 and the sum to infinity is 2, find the common ratio.

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

Correct answer: 3/4

S=a/(1-r)=2 so 1/2/(1-r)=2, giving 1-r=1/4, r=3/4.

Question 45: If z = 1 + i, what is z^2?

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

Correct answer: 2i

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

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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