AIME Inequalities and Optimization 3

Question 1

For positive reals x and y with xy = 1, what is the minimum value of (x + 1/x)² + (y + 1/y)²?

Accepted Answer

8

Answer

Since y = 1/x, both terms equal (x + 1/x)², giving 2(x + 1/x)² ≥ 2·(2)² = 8 by AM-GM.

Question 2

Among all triangles with perimeter 12, what is the maximum area?

Accepted Answer

4√3

Answer

The equilateral triangle maximizes area for a fixed perimeter; with side length 4, area = (√3/4)·16 = 4√3.

Question 3

For positive reals a and b satisfying a + b = 1, what is the minimum value of a³ + b³?

Accepted Answer

1/4

Answer

a³+b³ = (a+b)((a+b)²−3ab) = 1−3ab ≥ 1−3/4 = 1/4, since ab ≤ (a+b)²/4 = 1/4.

Question 4

What is the maximum value of 2x + y for real x, y satisfying x² + y² ≤ 5?

Accepted Answer

5

Answer

By Cauchy-Schwarz, (2x+y)² ≤ (4+1)(x²+y²) ≤ 5·5 = 25, so the maximum is 5.

Question 5

For non-negative integers m and n with m + n = 10, what is the maximum value of m·n?

Accepted Answer

25

Answer

m·n = m(10−m) is maximized at m = 5, giving 5·5 = 25.

AIME - American Invitational Mathematics Examination Practice Test