Algebra, Inequalities, Polynomials and Functional Equations — Practice

Question 1

Find the minimum of x + 1/x for x > 0.

Accepted Answer

By AM-GM it is at least 2, attained at x = 1.

Question 2

The roots of x^3 - 6x^2 + 11x - 6 are alpha, beta, gamma. Find the sum of their squares.

Accepted Answer

36 - 2(11) = 14.

Question 3

Prove (a+b)(b+c)(c+a) is at least 8abc for positive reals.

Accepted Answer

By AM-GM each factor is at least twice the square root of the product of its terms, so the product is at least 8 root(a^2 b^2 c^2) = 8abc, with equality at a = b = c.

Question 4

If a + b + c = 6, find the minimum of a^2 + b^2 + c^2.

Accepted Answer

By Cauchy 3(a^2 + b^2 + c^2) is at least 36, so the minimum is 12, at a = b = c = 2.

Question 5

Solve a(n+1) = 2a(n) + 1 with a(1) = 1.

Accepted Answer

Adding 1 gives a(n+1) + 1 = 2(a(n) + 1), so a(n) + 1 = 2^n and a(n) = 2^n - 1.

Question 6

Find the minimum of x + 4/x^2 for x > 0.

Accepted Answer

Write it as x/2 + x/2 + 4/x^2. The product is 1, so the sum is at least 3, with equality at x = 2.

Question 7

If f(x + f(y)) = f(x) + y for all reals, show that f(0) = 0 and f(f(y)) = y.

Accepted Answer

Put x = 0 to get f(f(y)) = f(0) + y, so f is injective. Put y = 0 to get f(x + f(0)) = f(x), so f(0) = 0 by injectivity. Then f(f(y)) = y.

Question 8

What is the remainder when P(x) is divided by x - 3?

Accepted Answer

P(3).

Practice — Algebra, Inequalities, Polynomials and Functional Equations

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