AIME Inequalities and Optimization 1 — Questions and Answers
Question 1: For a positive real number x, what is the minimum value of x + 4/x?
- 2
- 4 (Correct answer)
- 3
- 6
Correct answer: 4
By AM-GM, x + 4/x ≥ 2√(x · 4/x) = 2√4 = 4, with equality when x = 2.
Question 2: If a and b are positive reals with a + b = 10, what is the maximum value of ab?
- 20
- 24
- 25 (Correct answer)
- 30
Correct answer: 25
By AM-GM, ab ≤ ((a+b)/2)² = 25, with equality when a = b = 5.
Question 3: For real numbers x and y satisfying x + y = 6, what is the minimum value of x² + y²?
- 16
- 18 (Correct answer)
- 20
- 24
Correct answer: 18
Maximizing xy (by AM-GM, xy ≤ 9) minimizes x² + y² = (x+y)² − 2xy = 36 − 18 = 18.
Question 4: For positive reals x, y, z with x + y + z = 9, what is the maximum value of xy + yz + xz?
- 21
- 24
- 27 (Correct answer)
- 30
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 5: For positive reals a, b, c with abc = 1, what is the minimum value of a + b + c?
- 1
- 2
- 3 (Correct answer)
- 4
Correct answer: 3
By AM-GM, a + b + c ≥ 3·(abc)^(1/3) = 3·1 = 3, with equality when a = b = c = 1.
Question 6: For positive reals a and b, what is the minimum value of (a² + b²) / (ab)?
- 1
- 2 (Correct answer)
- 3
- 4
Correct answer: 2
(a² + b²)/(ab) = a/b + b/a ≥ 2 by AM-GM, with equality when a = b.
Question 7: What is the maximum value of 3x − 4y for real numbers x and y satisfying x² + y² = 25?
- 15
- 20
- 25 (Correct answer)
- 30
Correct answer: 25
By Cauchy-Schwarz, (3x − 4y)² ≤ (3² + 4²)(x² + y²) = 25·25 = 625, so the maximum is 25.
AIME Inequalities and Optimization 1
For a positive real number x, what is the minimum value of x + 4/x?