American Invitational Mathematics Examination (AIME) — Questions and Answers

Question 1: What is the sum of all positive integers less than 50 that are divisible by 7?

245
168 (Correct answer)
98
140

Correct answer: 168

The integers are 7, 14, 21, 28, 35, and 42. Their sum is 7+14+21+28+35+42= 147.

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

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

Correct answer: 2sin(θ)cos(θ)

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

Question 3: How many solutions does 2cos(x) = 1 have in the interval [0, 2π)?

4
2 (Correct answer)
0
1

Correct answer: 2

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

Question 4: 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)
(x+1)(x-2)(x+3) (Correct answer)

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

1326 (Correct answer)
1348
1378
1324

Correct answer: 1326

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

Question 6: Two parallel lines are cut by a transversal. If one interior angle is 65°, what is the co-interior (same-side interior) angle?

55°
115° (Correct answer)
125°
65°

Correct answer: 115°

Co-interior angles are supplementary, so 180°-65°=115°.

Question 7: What is the period of f(x) = sin(3x)?

3π
2π
2π/3 (Correct answer)
6π

Correct answer: 2π/3

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

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

3
2
1
0 (Correct answer)

Correct answer: 0

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

Question 9: In the Fibonacci sequence 1,1,2,3,5,8,13,…, what is the 9th term?

34 (Correct answer)
55
13
21

Correct answer: 34

Continuing: 1,1,2,3,5,8,13,21,34; the 9th term is 34.

Question 10: Two similar triangles have corresponding sides in ratio 3:5. What is the ratio of their areas?

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

Correct answer: 9:25

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

Question 11: For positive reals a and b satisfying a + b = 1, what is the minimum value of a³ + b³?

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

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 12: For positive reals a, b, c with a + b + c = 1, what is the minimum value of a² + b² + c²?

1/2
1/6
1/3 (Correct answer)
1/4

Correct answer: 1/3

By Cauchy-Schwarz, (a²+b²+c²)(1+1+1) ≥ (a+b+c)² = 1, so a²+b²+c² ≥ 1/3, with equality when a=b=c=1/3.

Question 13: For positive reals x, y, z with x + y + z = 9, what is the maximum value of xy + yz + xz?

24
30
21
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 14: What is the area of an equilateral triangle with side length 4?

8√3
2√3
16√3
4√3 (Correct answer)

Correct answer: 4√3

Area = (√3/4)×4^2=(√3/4)×16=4√3.

Question 15: In triangle ABC, angle A = 50° and angle B = 70°. What is angle C?

70°
50°
80°
60° (Correct answer)

Correct answer: 60°

Angles sum to 180°: C=180-50-70=60°.

Question 16: What is the argument (angle) of the complex number z = -1 + i?

45°
135° (Correct answer)
315°
225°

Correct answer: 135°

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

Question 17: Find the 10th term of the arithmetic sequence 3, 7, 11, 15, …

43
41
39 (Correct answer)
35

Correct answer: 39

a_n=3+(n-1)×4; a_10=3+36=39.

Question 18: Compute the conjugate of z = 7 - 2i.

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

Correct answer: 7 + 2i

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

Question 19: Find the number of terms in the arithmetic sequence 5, 9, 13, …, 101.

20
30
25 (Correct answer)
24

Correct answer: 25

101=5+(n-1)×4 gives n-1=24, so n=25.

Question 20: What is the maximum value of 2x + y for real x, y satisfying x² + y² ≤ 5?

5 (Correct answer)
5√2
4
2√5

Correct answer: 5

By Cauchy-Schwarz, (2x+y)² ≤ (4+1)(x²+y²) ≤ 5·5 = 25, so the maximum is 5.

Question 21: For positive reals a and b, what is the minimum value of (a² + b²) / (ab)?

3
1
2 (Correct answer)
4

Correct answer: 2

(a² + b²)/(ab) = a/b + b/a ≥ 2 by AM-GM, with equality when a = b.

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

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

Correct answer: 2i

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

Question 23: For a positive real number x, what is the minimum value of x + 4/x?

2
6
3
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 24: In triangle ABC with sides a=7, b=24, c=25, what type of triangle is it?

Equilateral
Obtuse
Acute
Right (Correct answer)

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 25: Compute (2+3i)(1-i).

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

Correct answer: 5+i

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

Question 26: The diagonals of a rhombus are 10 and 24. Find its perimeter.

48
56
60
52 (Correct answer)

Correct answer: 52

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

Question 27: What is the sum of interior angles of a hexagon?

720° (Correct answer)
540°
360°
900°

Correct answer: 720°

Sum = (n-2)×180°=(6-2)×180°=720°.

Question 28: 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 29: What is the largest integer n such that n^2 + 4n < 45?

6
5 (Correct answer)
4
7

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

8π
4π
16π (Correct answer)
12π

Correct answer: 16π

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

Question 31: 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 32: A sphere has radius 3. Find its volume in terms of π.

108π
27π
12π
36π (Correct answer)

Correct answer: 36π

V=(4/3)πr^3=(4/3)π(27)=36π.

Question 33: Find the value of (1 + √2)^4 - (1 - √2)^4.

24
16√2 (Correct answer)
12√2
8√2

Correct answer: 16√2

Expanding via binomial theorem, the rational terms cancel and irrational terms add: result is 2(4·√2+4·√2)... careful expansion gives 16√2.

Question 34: Simplify (2+i)/(1-i).

(3-i)/2
(1-3i)/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 35: A cone has base radius 3 and height 4. Find its slant height.

7
5 (Correct answer)
√7
√25

Correct answer: 5

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

Question 36: How many terms of the series 1+3+5+7+… are needed to reach a sum of 81?

7
9 (Correct answer)
8
11

Correct answer: 9

Sum of first n odd numbers=n^2; n^2=81 gives n=9.

Question 37: For positive reals a, b, c with abc = 1, what is the minimum value of a + b + c?

3 (Correct answer)
2
4
1

Correct answer: 3

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

Question 38: What is i^10 (where i = √(-1))?

-1 (Correct answer)
i
1
-i

Correct answer: -1

i^10=(i^4)^2·i^2=1·(-1)=-1.

Question 39: 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 40: A cylinder has radius 5 and height 8. What is its total surface area?

130π (Correct answer)
80π
160π
100π

Correct answer: 130π

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

Question 41: Find the modulus of the complex number z = 3 + 4i.

1
√7
5 (Correct answer)
7

Correct answer: 5

|z|=√(3^2+4^2)=√(9+16)=√25=5.

Question 42: Chord AB and chord CD intersect inside a circle. If AX=3, XB=8, CX=4, find XD.

8
6 (Correct answer)
12
4

Correct answer: 6

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

Question 43: 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 44: Find all values of x where f(x) = x^3 - 3x is increasing.

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

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 45: How many real roots does x^4 + 4 = 0 have?

1
0 (Correct answer)
2
4

Correct answer: 0

x^4=-4 has no real solutions since x^4≥0 for all real x.

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%

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