Advanced Topics · Chapter 40

Inequalities

Squares are never negative, averages beat geometric means, and dot products never beat lengths: three inequalities that find every maximum and minimum on the AMC.

What's this chapter about?

40.1 Trivial Inequality
40.2 AM-GM
40.3 Cauchy-Schwarz
40.4 Practice Problems
40.5 Additional Problems

You'll need

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40.10%

Trivial Inequality

In this section you will be able to use the fact that squares are never negative to find the lowest (or highest) value of a quadratic.

$(xy - 1)^2 + (x + y)^2$ can’t be negative: it’s a sum of squares. But how small can it actually get?

Every inequality in this chapter grows out of one fact: a real number squared is never negative.

To find the lowest point of a quadratic, complete the square. Write it as a square plus a constant. The square is at least $0$ , so the constant is the minimum, and it happens when the square is $0$ .

The trick also works on bigger expressions. Look for a substitution that turns the expression into “something squared, plus a number”.

Exploreplay first, formulas later

Degree2

Lead a1

P(x) = (x^{2} - 6x + 13) = x^{2} - 6x + 13

P(x) = ax^2 + bx + c = a(x - r_1)\cdots(x - r_2)

r_1, r_2 = 3 \pm 2i

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

vertex

(h, k) = (3, 4),\; P(x) = (x - 3)^2 + 4

k = c - \frac{b^2}{4a}

lowest point

P(x) \ge 4 \text{ at } x \approx 3

Try this: Drag the parabola's vertex. Which coefficient controls its height?
Try this: Read the completed-square form. Why can the graph never dip below the vertex?
Try this: Look at the Why? tiles under Corollary 40.1.2. Fill the missing corner: what did you add?

Name it

Theorem 40.1.1 · Trivial Inequality

For real $x$ ,

x^2 \ge 0.

This means all perfect squares are $0$ or more.

Corollary 40.1.2 · Completing the Square

In a quadratic $Q(x) = ax^2 + bx + c$ ,

If $a > 0$ , then the minimum value of $Q(x)$ is $c - \frac{b^2}{4a}$ and occurs when $x = -\frac{b}{2a}$ .
If $a < 0$ , then the maximum value of $Q(x)$ is $c - \frac{b^2}{4a}$ and occurs when $x = -\frac{b}{2a}$ .

Try it

Problem 40.4.3

AMC 10/12

What is the least possible value of $(xy - 1)^2 + (x + y)^2$ for real numbers $x$ and $y$ ?

Problem 40.4.2

AMC 10

What is the least possible value of

(x + 1)(x + 2)(x + 3)(x + 4) + 2019

where $x$ is a real number?

Degree4

Lead a1

P(x) = (x + 4)(x + 3)(x + 2)(x + 1) = x^{4} + 10x^{3} + 35x^{2} + 50x + 24

r_1 = -4,\; r_2 = -3,\; r_3 = -2,\; r_4 = -1

lowest point

P(x) \ge ? \text{ at } x \approx ?

Contest cornerreal contest problems

Problem 40.4.1

AMC 10

Define

P(x) = (x - 1^2)(x - 2^2) \cdots (x - 100^2).

How many integers $n$ are there such that $P(n) \le 0$ ?

Checkpoint for section 40.1

Get 2 of 3 right to master this section. One try each, no hints.

40.2AMC 120%

AM-GM

In this section you will be able to use AM-GM to make products as big as possible and sums as small as possible, and check when equality holds.

$A + M + C = 12$ . How big can you make $AMC + AM + MC + CA$ ? Hint: equal parts win.

Look:From here on, the inequalities are optional for the AMC 10 but common on the AMC 12 and AIME.

The arithmetic mean of non-negative numbers is never smaller than their geometric mean. With a fixed sum, the product is biggest when all the numbers are equal. With a fixed product, the sum is smallest when they are equal.

The hard part is setting it up. You need a constant sum (to bound a product) or a constant product (to bound a sum). If the terms don’t cooperate, multiply some of them by weights until they do, then divide back at the end.

Always check the equality case: the bound only counts if equal parts are actually allowed.

Exploreplay first, formulas later

Try this: Press Make it a square. Can any other rectangle with the same perimeter beat its area?
Try this: In Example 40.1's split, 15 is cut into 3 whole parts. Which split wins?
Try this: Ex 40.2: why multiply 4 − x by 3 before applying AM-GM?

Name it

Theorem 40.2.1 · AM-GM Inequality For 2 variables

For non-negative reals $a$ and $b$ ,

\frac{a+b}{2} \ge \sqrt{ab}.

Basically, this means the average of 2 non-negative numbers (arithmetic mean) is always at least as big as the square root of the product of the 2 numbers (the geometric mean).

Equality holds when $a = b$ .

Corollary 40.2.2

For $x > 0$ , the minimum value of $x + \frac{1}{x}$ is $2$ and occurs when $x = 1$ .

The minimum value of $a + b$ (if $ab$ remains constant) occurs when $a = b$ .
The maximum value of $ab$ (if $a + b$ remains constant) occurs when $a = b$ .

Theorem 40.2.3 · AM-GM Inequality For More Variables

For non-negative reals $a_1, a_2, \ldots, a_n$ ,

\frac{a_1 + a_2 + \cdots + a_n}{n} \ge \sqrt[n]{a_1 \cdot a_2 \cdot a_3 \cdots a_n}.

Equality occurs when $a_1 = a_2 = \cdots = a_n$ (all the variables are equal).

Another way to say this is

\text{Average of } n \text{ numbers} \ge \sqrt[n]{\text{product of all } n \text{ numbers}}.

Remark 40.2.5

Generally, we use AM-GM to maximize products or minimize sums.

Theorem 40.2.6 · Weighted AM-GM Inequality

For non-negative reals $a_i$ , $c_i$ ,

\frac{c_1 a_1 + c_2 a_2 + \cdots + c_n a_n}{c_1 + c_2 + \cdots + c_n} \ge \sqrt[c_1 + c_2 + \cdots + c_n]{a_1^{c_1} \cdot a_2^{c_2} \cdot a_3^{c_3} \cdots a_n^{c_n}}.

Remark 40.2.8

We use weighted AM-GM when we are trying to make the sum of all terms a constant by multiplying weights to all (or some) of the terms. Remember to divide by the weights you multiplied at the end.

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Example 40.1

AMC 12

Let $A$ , $M$ , and $C$ be non-negative integers such that $A + M + C = 12$ . What is the maximum value of $A \cdot M \cdot C + A \cdot M + M \cdot C + A \cdot C$ ?

1
$AMC + AM + MC + CA$ looks like part of an expanded product. Expand $(A+1)(M+1)(C+1)$ . What extra terms appear?

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Example 40.2

Find the maximum value of $x^3(4 - x)$ .

1
Think of $x^3(4-x)$ as the product $x \cdot x \cdot x \cdot (4 - x)$ . Is the sum $x + x + x + (4-x)$ constant?

Predict first:
Click a choice or type its letter (A–E).

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Example 40.3

AMC 12

What is the maximum value of $\frac{(2^t - 3t)t}{4^t}$ for real values of $t$ ?

1
Split the fraction: $\frac{(2^t - 3t)t}{4^t} = \frac{t}{2^t} - \frac{3t^2}{4^t}$ . Let $u = \frac{t}{2^t}$ . Write the expression in terms of $u$ .

Predict first:
Your prediction

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Try it

Problem 40.4.6

AMC 12

How many ordered pairs of positive real numbers $(a, b)$ satisfy the equation

(1 + 2a)(2 + 2b)(2a + b) = 32ab?

Contest cornerreal contest problems

Problem 40.4.8

AIMEChallenge

Let $\mathcal{B}$ be the set of rectangular prisms that have volume $23$ and surface area $54$ . Suppose $r$ is the least possible radius of a sphere that can fit any element of $\mathcal{B}$ inside it. Then $r^2$ can be expressed as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$ .

Problem 40.5.4(additional)

AMC 12

There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3 - ax^2 + bx - a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is this value of $b$ ?

Problem 40.5.5(additional)

AIME

Let $x_1, x_2, \ldots, x_6$ be non-negative real numbers such that $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1$ , and $x_1x_3x_5 + x_2x_4x_6 \ge \frac{1}{540}$ . Let $p$ and $q$ be positive relatively prime integers such that $\frac{p}{q}$ is the maximum possible value of $x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2$ . Find $p + q$ .

Checkpoint for section 40.2

Get 2 of 3 right to master this section. One try each, no hints.

40.3AMC 120%

Cauchy-Schwarz

In this section you will be able to use Cauchy-Schwarz (and Titu's Lemma) to link a sum with a sum of squares.

$a + b + c = 2$ and $a^2 + b^2 + c^2 = 12$ . How far apart can the largest and smallest possible $c$ be?

Think of $(a_1, a_2)$ and $(b_1, b_2)$ as arrows. Their dot product $a_1b_1 + a_2b_2$ is the product of their lengths times $\cos\theta$ . Since $\cos^2\theta \le 1$ , the squared dot product can’t beat the product of the squared lengths.

That is Cauchy-Schwarz. It turns a plain sum like $a + b$ into a sum of squares: pair $(a, b)$ with $(1, 1)$ . Equality happens when the arrows point the same way, that is when the ratios match.

Finding an extreme value always takes two steps: prove the bound, then show some numbers reach it.

Exploreplay first, formulas later

Try this: Drag b until the two bars meet. What is the angle between the arrows?
Try this: The gap is |a|²|b|² sin²θ. Why can it never be negative?
Try this: Ex 40.4: which vector pairs with (a, b) to bring in a + b?

Name it

Theorem 40.3.1 · Cauchy-Schwarz

For reals $a_i$ and $b_i$ ,

(a_1 b_1 + a_2 b_2 + \cdots + a_n b_n)^2 \le \left(a_1^2 + a_2^2 + \cdots + a_n^2\right)\left(b_1^2 + b_2^2 + \cdots + b_n^2\right).

This means: (the sum of the products of all $a_k$ and $b_k$ ) $^2$ $\le$ the product of the sums of squares of all $a_k$ and all $b_k$ .

Equality holds when the ratio $\frac{a_i}{b_i}$ is the same for all $i$ .

Remark 40.3.3

You generally want to apply Cauchy-Schwarz when you are dealing with sums of squares.

Corollary 40.3.4 · Titu's Lemma

For reals $a_i$ and positive reals $b_i$ ,

\frac{a_1^2}{b_1} + \frac{a_2^2}{b_2} + \cdots + \frac{a_n^2}{b_n} \ge \frac{(a_1 + a_2 + \cdots + a_n)^2}{b_1 + b_2 + \cdots + b_n}.

Alternately (for non-negative $a_i$ ),

\frac{a_1}{b_1} + \frac{a_2}{b_2} + \cdots + \frac{a_n}{b_n} \ge \frac{\left(\sqrt{a_1} + \sqrt{a_2} + \cdots + \sqrt{a_n}\right)^2}{b_1 + b_2 + \cdots + b_n}.

This is a direct consequence of Cauchy-Schwarz.

Concept 40.3.5 · Techniques For Optimization Problems

Steps to find the maximum/minimum of expressions:

Find another simple expression that is greater than or equal to the given expression (for a maximization) or less than or equal to it (for a minimization), using one (or possibly more) of these inequalities:
- Trivial Inequality
- AM-GM
- Weighted AM-GM (AM-GM, weighted and unweighted, is useful for maximizing products and minimizing sums)
- Cauchy-Schwarz (useful when dealing with sums of squares)
Verify that the equality case of your inequality can hold under your problem's conditions.
Simplify your equality case and solve for the answer.

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Example 40.4

AMC 12

Let $a$ , $b$ , and $c$ be real numbers such that

a + b + c = 2, \quad \text{and} \quad a^2 + b^2 + c^2 = 12.

What is the difference between the maximum and minimum possible values of $c$ ?

1
Treat $c$ as fixed and move it across. What is $a + b$ in terms of $c$ ?

Predict first:
Your prediction

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Contest cornerreal contest problems

Problem 40.5.2(additional)

AIME

Suppose that $|x_i| < 1$ for $i = 1, 2, \ldots, n$ . Suppose further that $|x_1| + |x_2| + \cdots + |x_n| = 19 + |x_1 + x_2 + \cdots + x_n|$ . What is the smallest possible value of $n$ ?

Problem 40.5.3(additional)

AIME

This problem also appears in the book as 12.6.16.

Let $x_1 \le x_2 \le \cdots \le x_{100}$ be real numbers such that $|x_1| + |x_2| + \cdots + |x_{100}| = 1$ and $x_1 + x_2 + \cdots + x_{100} = 0$ . Among all such 100-tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Checkpoint for section 40.3

Get 2 of 3 right to master this section. One try each, no hints.

40.4

Practice Problems

In this section you will be able to bound expressions and count solutions of inequality systems.

These practice problems mix inequalities with counting and functions. Each one has three hints if you get stuck.

The other practice problems (40.4.1–40.4.3, 40.4.6, 40.4.8) sit in the lessons they use. Want them all on one page? Open the problem set.

Contest cornerreal contest problems

Problem 40.4.4

AMC 10

How many ordered triples of integers $(x, y, z)$ satisfy the following system of inequalities?

\begin{aligned} -x - y - z &\le -2 \\ -x + y + z &\le 2 \\ x - y + z &\le 2 \\ x + y - z &\le 2 \end{aligned}

Problem 40.4.5

AMC 10

Consider functions $f$ that satisfy $|f(x) - f(y)| \le \frac12 |x - y|$ for all real numbers $x$ and $y$ . Of all such functions that also satisfy the equation $f(300) = f(900)$ , what is the greatest possible value of

f(f(800)) - f(f(400))?

Problem 40.4.7

AMC 12

A real-valued function $f$ has the property that for all real numbers $a$ and $b$ ,

f(a + b) + f(a - b) = 2f(a)f(b).

Which one of the following cannot be the value of $f(1)$ ?

40.5

Additional Problems

In this section you will be able to mix bounding and equality cases on harder contest problems.

The additional problems. Most are AIME: first find a bound, then build numbers that reach it.

Problems 40.5.2–40.5.5 sit in the lessons they use (AM-GM and Cauchy-Schwarz).

Contest cornerreal contest problems

Problem 40.5.1(additional)

AMC 10/12

This problem also appears in the book as 40.4.3.

What is the least possible value of $(xy - 1)^2 + (x + y)^2$ for real numbers $x$ and $y$ ?

Problem 40.5.6(additional)

AIME

Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta, who did not attempt 300 points on the first day, had a positive integer score on each of the two days, and Beta's daily success rate (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was $\frac{300}{500} = \frac35$ . The largest possible two-day success ratio that Beta could achieve is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m + n$ ?