AIME Inequalities and Optimization 2

Question 1

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

Accepted Answer

4

Answer

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?

Accepted Answer

√2

Answer

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?

Accepted Answer

1/3

Answer

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)?

Accepted Answer

4

Answer

(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?

Accepted Answer

7

Answer

Squaring x² + x⁻² = 3 gives x⁴ + 2 + x⁻⁴ = 9, so x⁴ + x⁻⁴ = 7.

AIME - American Invitational Mathematics Examination Practice Test