American Invitational Mathematics Examination (AIME) — Questions and Answers

Question 1: What is sin(30°) + cos(60°)?

1/2
√3/2
1 (Correct answer)
√2

Correct answer: 1

sin(30°)=1/2 and cos(60°)=1/2; sum=1.

Question 2: A rectangle has perimeter 36 and width 6. What is its area?

72 (Correct answer)
54
36
90

Correct answer: 72

Length=(36-12)/2=12; area=12×6=72.

Question 3: How many real roots does x^4 + 4 = 0 have?

0 (Correct answer)
1
2
4

Correct answer: 0

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

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

4
7
5 (Correct answer)
6

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

10
5√2
5√3
5 (Correct answer)

Correct answer: 5

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

Question 6: How many solutions does 2cos(x) = 1 have in the interval [0, 2π)?

1
2 (Correct answer)
4
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 7: What is the value of tan(45°)?

1 (Correct answer)
√3
1/√3
0

Correct answer: 1

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

Question 8: What is the maximum value of 2x + y for real x, y satisfying x² + y² ≤ 5?

5√2
4
5 (Correct answer)
2√5

Correct answer: 5

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

Question 9: Compute (2+3i)(1-i).

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

Correct answer: 5+i

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

Question 10: For positive reals x, y, z with x + y + z = 9, what is the maximum value of xy + yz + xz?

30
21
24
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 11: For what value of c does cx^2 + 8x + 4 = 0 have a double root?

4 (Correct answer)
8
2
1

Correct answer: 4

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

Question 12: For positive reals a and b satisfying a + b = 1, what is the minimum value of a³ + b³?

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

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

98
140
245
168 (Correct answer)

Correct answer: 168

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

Question 14: 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 15: 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 16: How many integer solutions does x^2 < 16 have?

8
7 (Correct answer)
4
6

Correct answer: 7

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

Question 17: If 𝑎 and 𝑏 are relatively prime, which of the following statements is true?

𝑎 + 𝑏 is always even
𝑎 ⋅ 𝑏 is always a perfect square
𝑎 and 𝑏 are both prime numbers
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 18: A cylinder has radius 5 and height 8. What is its total surface area?

80π
100π
160π
130π (Correct answer)

Correct answer: 130π

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

Question 19: In how many ways can you choose 2 different cards from a standard deck of 52 cards?

1326 (Correct answer)
1324
1348
1378

Correct answer: 1326

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

Question 20: The sum of an infinite geometric series is 12 and the first term is 3. Find the common ratio.

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

Correct answer: 3/4

12=3/(1-r), so 1-r=3/12=1/4, r=3/4.

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

8 (Correct answer)
23
28
13

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

3
2 (Correct answer)
4
1

Correct answer: 2

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

Question 23: 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 24: Chord AB and chord CD intersect inside a circle. If AX=3, XB=8, CX=4, find XD.

6 (Correct answer)
4
12
8

Correct answer: 6

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

Question 25: Find the value of (1 + √2)^4 - (1 - √2)^4.

12√2
16√2 (Correct answer)
24
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 26: 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 27: A sphere has radius 3. Find its volume in terms of π.

108π
36π (Correct answer)
12π
27π

Correct answer: 36π

V=(4/3)πr^3=(4/3)π(27)=36π.

Question 28: A circle has center (3, -1) and passes through (7, -1). What is its area?

4π
16π (Correct answer)
8π
12π

Correct answer: 16π

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

Question 29: A sector of a circle has radius 6 and central angle 60°. What is the arc length?

2π (Correct answer)
π
6π
3π

Correct answer: 2π

Arc length = (60/360)×2π×6=2π.

Question 30: What is sin^2(θ) + cos^2(θ) for any angle θ?

2
0
sin(2θ)
1 (Correct answer)

Correct answer: 1

This is the fundamental Pythagorean identity, which equals 1 for all θ.

Question 31: Simplify (2+i)/(1-i).

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

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

16√3
8√3
4√3 (Correct answer)
2√3

Correct answer: 4√3

Area = (√3/4)×4^2=(√3/4)×16=4√3.

Question 33: A 4-digit number is such that the sum of its digits is 20. How many such numbers are divisible by 9?

5 (Correct answer)
4
7
6

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

55°
125°
65°
115° (Correct answer)

Correct answer: 115°

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

Question 35: In triangle ABC with sides a=7, b=24, c=25, what type of triangle is it?

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

13 (Correct answer)
9
12
10

Correct answer: 13

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

Question 37: Find the value of the sum Σ(k=1 to 5) k(k+1).

70 (Correct answer)
45
60
55

Correct answer: 70

k=1:2, k=2:6, k=3:12, k=4:20, k=5:30; sum=2+6+12+20+30=70.

Question 38: 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 39: How many ways can 5 people be seated in a row?

120 (Correct answer)
24
720
60

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 40: 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 41: A cone has base radius 3 and height 4. Find its slant height.

7
√25
5 (Correct answer)
√7

Correct answer: 5

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

Question 42: An arithmetic sequence has first term 5 and common difference -3. Find the sum of the first 10 terms.

50
-95
85
-85 (Correct answer)

Correct answer: -85

S_10=(10/2)(2×5+9×(-3))=5(10-27)=5(-17)=-85.

Question 43: 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.

Question 44: The sum of the arithmetic series 1+3+5+…+99 equals:

1000
2500 (Correct answer)
2550
5050

Correct answer: 2500

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

Question 45: How many ways can you distribute 5 identical candies to 3 children so that each child gets at least one candy?

6
10 (Correct answer)
15
21

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.

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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