AIME Inequalities and Optimization 2 — Questions and Answers
Question 1: For positive reals x and y with x + y = 1, what is the minimum value of 1/x + 1/y?
- 2
- 3
- 4 (Correct answer)
- 6
Correct 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.
Question 2: What is the maximum value of sin(θ) + cos(θ) over all real θ?
- 1
- √2 (Correct answer)
- √3
- 2
Correct answer: √2
sin(θ) + cos(θ) = √2·sin(θ + π/4), which has maximum value √2.
Question 3: For positive reals a, b, c with a + b + c = 1, what is the minimum value of a² + b² + c²?
- 1/6
- 1/4
- 1/3 (Correct answer)
- 1/2
Correct answer: 1/3
By Cauchy-Schwarz, (a²+b²+c²)(1+1+1) ≥ (a+b+c)² = 1, so a²+b²+c² ≥ 1/3, with equality when a=b=c=1/3.
Question 4: For positive reals x and y, what is the minimum value of (x + y)(1/x + 1/y)?
- 2
- 3
- 4 (Correct answer)
- 5
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 5: If x > 0 and x² + x⁻² = 3, what is the value of x⁴ + x⁻⁴?
- 5
- 6
- 7 (Correct answer)
- 9
Correct answer: 7
Squaring x² + x⁻² = 3 gives x⁴ + 2 + x⁻⁴ = 9, so x⁴ + x⁻⁴ = 7.
Question 6: What is the maximum product of positive integers that sum to 10?
- 24
- 36 (Correct answer)
- 40
- 45
Correct answer: 36
The partition 3+3+4 = 10 gives product 3·3·4 = 36, which is optimal since using 3s and 2s maximizes the product.
Question 7: For real numbers a and b satisfying a² + b² = 4, what is the maximum value of a + 2b?
- 4
- 2√5 (Correct answer)
- 2√6
- 6
Correct answer: 2√5
By Cauchy-Schwarz, (a + 2b)² ≤ (1² + 2²)(a² + b²) = 5·4 = 20, so the maximum is √20 = 2√5.
AIME Inequalities and Optimization 2
For positive reals x and y with x + y = 1, what is the minimum value of 1/x + 1/y?