AIME Inequalities and Optimization

Question 1

For a positive real number x, what is the minimum value of x + 4/x?

Accepted Answer

4

Answer

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?

Accepted Answer

25

Answer

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?

Accepted Answer

18

Answer

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?

Accepted Answer

27

Answer

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?

Accepted Answer

3

Answer

By AM-GM, a + b + c ≥ 3·(abc)^(1/3) = 3·1 = 3, with equality when a = b = c = 1.

AIME - American Invitational Mathematics Examination Practice Test