AIME Algebra 3

Question 1

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

Accepted Answer

15

Answer

2^(a+b)=2^a · 2^b=3·5=15.

Question 2

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

Accepted Answer

5

Answer

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 3

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

Accepted Answer

16√2

Answer

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 4

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

Accepted Answer

3

Answer

Since f(f(x))=x, applying f to both sides of f(3)=7 gives f(f(3))=f(7), so f(7)=3.

Question 5

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

Accepted Answer

2500

Answer

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

AIME - American Invitational Mathematics Examination Practice Test

AIME - American Invitational Mathematics Examination Practice Test

AIME Algebra 3