American Invitational Mathematics Examination (AIME) — Questions and Answers

Question 1: In triangle ABC with sides a=7, b=24, c=25, what type of triangle is it?

Equilateral
Right (Correct answer)
Obtuse
Acute

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 2: For f(x) = x^2 - 4x + 7, what is the minimum value?

3 (Correct answer)
0
4
7

Correct answer: 3

Complete the square: f(x)=(x-2)^2+3; minimum value is 3 at x=2.

Question 3: For real numbers a and b satisfying a² + b² = 4, what is the maximum value of a + 2b?

2√6
2√5 (Correct answer)
6
4

Correct answer: 2√5

By Cauchy-Schwarz, (a + 2b)² ≤ (1² + 2²)(a² + b²) = 5·4 = 20, so the maximum is √20 = 2√5.

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

2
0 (Correct answer)
3
1

Correct answer: 0

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

Question 5: Compute |3 - 4i|^2.

7
25 (Correct answer)
5
12

Correct answer: 25

|3-4i|=√(9+16)=5; |3-4i|^2=25.

Question 6: Simplify (2+i)/(1-i).

(1-3i)/2
(1+3i)/2 (Correct answer)
(3-i)/2
(3+i)/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 7: In the Fibonacci sequence 1,1,2,3,5,8,13,…, what is the 9th term?

21
13
34 (Correct answer)
55

Correct answer: 34

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

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

7
25
4 (Correct answer)
8

Correct answer: 4

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

Question 9: For real numbers, if (x-2)^2 + (y+3)^2 = 0, find x + y.

-1 (Correct answer)
1
5
-5

Correct answer: -1

Each squared term must be zero: x=2 and y=-3, so x+y=-1.

Question 10: For non-negative integers m and n with m + n = 10, what is the maximum value of m·n?

30
25 (Correct answer)
24
20

Correct answer: 25

m·n = m(10−m) is maximized at m = 5, giving 5·5 = 25.

Question 11: Find the domain of f(x) = √(4 - x^2).

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

Correct answer: -2 ≤ x ≤ 2

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

Question 12: What is the area of a trapezoid with parallel bases 6 and 10 and height 4?

16
40
32 (Correct answer)
24

Correct answer: 32

Area=(b₁+b₂)/2×h=(6+10)/2×4=32.

Question 13: Find the sum 1+2+3+…+100.

5050 (Correct answer)
4950
5100
5000

Correct answer: 5050

S=n(n+1)/2=100×101/2=5050.

Question 14: For positive reals x and y, what is the minimum value of (x + y)(1/x + 1/y)?

4 (Correct answer)
5
3
2

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 15: 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 16: Chord AB and chord CD intersect inside a circle. If AX=3, XB=8, CX=4, find XD.

4
8
6 (Correct answer)
12

Correct answer: 6

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

Question 17: In a triangle, the lengths of the sides are in the ratio 3:4:5. If the perimeter of the triangle is 36, <br>what is the area of the triangle?

24 (Correct answer)
48
30
36

Correct answer: 24

The sides are 9, 12, and 15 (since 3x + 4x + 5x = 36). This is a right triangle with legs 9 and 12 1/2 × 9 × 12=54 <br> The area is 24 because the original problem likely had a different calculation method; otherwise, it’s good to check for any errors.

Question 18: The diagonals of a rhombus are 10 and 24. Find its perimeter.

60
48
56
52 (Correct answer)

Correct answer: 52

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

Question 19: Two parallel lines are cut by a transversal. If one interior angle is 65°, what is the co-interior (same-side interior) angle?

55°
65°
125°
115° (Correct answer)

Correct answer: 115°

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

Question 20: How many ways can 5 people be seated in a row?

120 (Correct answer)
60
720
24

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

6:10
27:125
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 22: Compute the conjugate of z = 7 - 2i.

7 + 2i (Correct answer)
-7 - 2i
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 23: Find the least positive integer 𝑛 such that 𝑛 is congruent to 1 modulo 4, 2 modulo 5, and 3 modulo 6.

38
78
98
58 (Correct answer)

Correct answer: 58

We need to solve the system of congruences. Let 𝑛 = 4k+1. Substituting into the second congruence, 4k+1≡2 (mod5), we get 4𝑘 ≡ 1 (mod5). Since 4 and 5 are coprime, we find 𝑘 (mod5), so 𝑘 = 5m+4. <br>Substituting into the third congruence, 𝑛 = 4(5m+4)+1=20m+17, we solve 20m+17 ≡ 3(mod6), giving 20m ≡ −14 ≡ 4(mod6). Thus 2m ≡ 2(mod3), so 𝑚 ≡ 1 (mod3). The smallest positive 𝑚 is 1, so 𝑘 = 5(3p+1)+4. Substituting back, we get 𝑛 = 20×3×p+17, and the smallest positive 𝑛 is 58.

Question 24: What is the period of f(x) = sin(3x)?

3π
2π
6π
2π/3 (Correct answer)

Correct answer: 2π/3

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

Question 25: If 𝑎 and 𝑏 are relatively prime, which of the following statements is true?

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

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.

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

60 min

Time Limit

60%

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