Mixed Deck — All AIME Topics Flashcards

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

Answer: 34

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

If a + b = 5 and ab = 3, what is a^3 + b^3?

Answer: 80

Using the identity a^3+b^3=(a+b)^3-3ab(a+b)=125-45=80.

The roots of 3x^2 - 5x + 1 = 0 are r and s. Find r^2 + s^2.

Answer: 19/9

r+s=5/3, rs=1/3; r^2+s^2=(r+s)^2-2rs=25/9-2/3=25/9-6/9=19/9.

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

Answer: 39

a_n=3+(n-1)×4; a_10=3+36=39.

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

Answer: 168

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

For positive reals x and y with x + y = 1, what is the minimum value of 1/x + 1/y?

Answer: 4

1/x + 1/y = (x+y)/(xy) = 1/(xy) ≥ 4 since xy ≤ (x+y)²/4 = 1/4, with equality when x = y = 1/2.

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

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.

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

Answer: 3/4

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

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

Answer: 9:25

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

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

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.)

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

Answer: 7

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

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

Answer: 3

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

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

Answer: 2i

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

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

Answer: 32

Area=(b₁+b₂)/2×h=(6+10)/2×4=32.

Find the common ratio of the geometric sequence 3, 6, 12, 24, …

Answer: 2

Each term is doubled: ratio = 6/3 = 2.

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

Answer: -85

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

Which of the following correctly states the AM-GM inequality for three positive reals x, y, z?

Answer: (x+y+z)/3 ≥ (xyz)^(1/3)

AM-GM states the arithmetic mean (x+y+z)/3 is always ≥ the geometric mean (xyz)^(1/3), with equality iff x=y=z.

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

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.

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

Answer: 52

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

If f(f(x)) = x for all x and f(3) = 7, find f(7).

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.