Practice Test GeeksAIME - American Invitational Mathematics Examination Practice Test
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Algebra Flashcards

6 cards from real AIME practice questions. Tap to flip, then mark Knew It or Still Learning — missed cards come back until you master them.

Read the first 6 Algebra flashcards as text

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

    Answer: 15

    2^(a+b)=2^a · 2^b=3·5=15.

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

    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.

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

    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.

  4. 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.

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

    Answer: 2500

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

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

    Answer: -1

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