Step-by-Step Adjustment

In one sentence

With many variables, thinking about all of them at once is too hard. Step-by-step adjustment says: freeze the other variables, move just two, and see which move makes the target better.

x_{i} + x_{j} fixed, ∣ x_{i} - x_{j} ∣ ↓ ⟹ x_{i}^{2} + x_{j}^{2} ↓, x_{i} x_{j} ↑

This is because $x_{i}^{2} + x_{j}^{2} = \frac{1}{2} [(x_{i} + x_{j})^{2} + (x_{i} - x_{j})^{2}]$ and $x_{i} x_{j} = \frac{1}{4} [(x_{i} + x_{j})^{2} - (x_{i} - x_{j})^{2}]$ . As long as two numbers are “not close enough”, you can improve one more step; the state where no further adjustment is possible is the optimal one.

Predict first

$4$ positive integers add up to $20$ . What is the smallest possible sum of their squares?

Choose an answer first; then you can see the result.

Example: x₁ + x₂ + x₃ + x₄ = 20 (positive integers). Find the minimum of the sum of squares

Step 1 / 4

Move only two
Pick the largest, $9$ , and the smallest, $1$ , and move each one step toward the middle: $9 \to 8$ , $1 \to 2$ . The sum does not change.

$(a - 1)^{2} + (b + 1)^{2} = a^{2} + b^{2} - 2 (a - b - 1)$
How much better?
Keep adjusting
Where it stops

Stop 1 · Proofs of the mean inequalities (Vol. 6)

Adjustment gives one of the most intuitive proofs of the mean inequality. The key is to make the process end after finitely many steps.

Example: positive numbers x₁, …, xₙ have sum nA. Then x₁x₂⋯xₙ ≤ Aⁿ

Step 1 / 4

If they are not all equal to A
Then the largest $M > A$ and the smallest $m < A$ .

$m < A < M$
One adjustment
A larger product, and one more A
At most n − 1 steps

Stop 2 · Extreme values: splitting a number for the largest product (Vol. 6)

When the number of terms is not fixed, an adjustment is not only “move two numbers closer”: you can also “split” or “merge”.

Four positive integers with sum

20

: moving them closer makes the product larger, up to

5 \times 5 \times 5 \times 5 = 625

. But if the number of terms is free, can it get even larger?

Example: write 20 as a sum of positive integers. What is the largest possible product?

Step 1 / 4

Merge any 1
$1 \cdot x < 1 + x$ : merging a $1$ into another number makes the product larger. So the optimum has no $1$ .

$1 \cdot x \to x + 1$
Split anything above 4
At most two 2s
The answer

Stop 3 · Trigonometric inequalities (Vol. 3)

One problem, several solutions

In $△ A B C$ , prove that $sin A + sin B + sin C \leq \frac{3 3}{2}$ .

$sin A + sin B = 2 sin \frac{A + B}{2} cos \frac{A - B}{2}$ : with $A + B$ fixed, the smaller $∣ A - B ∣$ is, the larger the sum. If the three angles are not all $6 0^{\circ}$ , then (after relabeling) $A < 6 0^{\circ} < B$ . Replace $A, B$ by $6 0^{\circ}$ and $A + B - 6 0^{\circ}$ : the sum of the angles does not change, and $∣6 0^{\circ} - (A + B - 6 0^{\circ}) ∣ < B - A$ , so $sin A + sin B$ increases. Now one angle is $6 0^{\circ}$ and the other two add up to $12 0^{\circ}$ ; one more adjustment makes them all $6 0^{\circ}$ . So the maximum is reached by the equilateral triangle: $3 sin 6 0^{\circ} = \frac{3 3}{2}$ .

CostUses only sum-to-product, and reaches the equilateral triangle in at most two steps.

Jensen's inequality

$sin x$ is concave on $(0, π)$ , so $\frac{sin A + sin B + sin C}{3} \leq sin \frac{A + B + C}{3} = sin 6 0^{\circ} = \frac{3}{2}$ .

CostOne line, but you must first know that sin x is concave on (0, π).

Stop 4 · Ordering and swapping (Vol. 1)

“Take the best arrangement; if a swap could still improve it…”: adjustment and the extremal principle often appear together.

Example: a₁ ≤ a₂ ≤ ⋯ ≤ aₙ and b₁ ≤ ⋯ ≤ bₙ. Over all permutations of the b’s, when is Σ aᵢb_{σ(i)} largest?

Step 1 / 4

Take a largest one
There are only finitely many permutations, so some $σ$ makes the sum largest.

$S (σ) = \sum a_{i} b_{σ (i)}$
Find an “inversion”
The swap does not hurt
Adjust until the orders match

Practice

Exercise 1Basic

$3$ positive integers add up to $20$ . What is the smallest possible sum of their squares?

Exercise 2Basic

$5$ positive integers add up to $23$ . What is the smallest possible sum of their squares?

Exercise 3Advanced

Some positive integers (any number of them) add up to $20$ . What is the largest possible product?

Exercise 4Challenge

Some positive integers add up to $2026$ . When their product is as large as possible, it equals $2^{a} \cdot 3^{b}$ . Find $a + b$ .

Step-by-Step Adjustment

In one sentence

Example: x₁ + x₂ + x₃ + x₄ = 20 (positive integers). Find the minimum of the sum of squares

Stop 1 · Proofs of the mean inequalities (Vol. 6)

Example: positive numbers x₁, …, xₙ have sum nA. Then x₁x₂⋯xₙ ≤ Aⁿ

Stop 2 · Extreme values: splitting a number for the largest product (Vol. 6)

Example: write 20 as a sum of positive integers. What is the largest possible product?

Stop 3 · Trigonometric inequalities (Vol. 3)

Step-by-step adjustment

Jensen's inequality

Stop 4 · Ordering and swapping (Vol. 1)

Example: a₁ ≤ a₂ ≤ ⋯ ≤ aₙ and b₁ ≤ ⋯ ≤ bₙ. Over all permutations of the b’s, when is Σ aᵢb_{σ(i)} largest?

Practice

Mistake Museum · Exhibits from this stop

An adjustment with no end

Pushing integers toward “all equal” too

Stops on this line