Book 2. Olympiad Inequalities

Key idea

The basic olympiad principle for inequalities is this: before estimating an expression, understand where equality could occur and which quantity is the right one to compare. Many solutions use no heavy theorem; the decisive step is choosing a square, an ordering, or an extremal element.

Basic facts

We shall use \(t^2\ge0\), \((x-y)^2\ge0\), ordering of numbers, the minimal sum of distinct positive integers, sign checks before multiplying inequalities, and equality-case analysis. Remember: an inequality may be multiplied by an expression only after its sign is controlled.

When to use this method

Use these principles when the problem contains words such as “largest”, “smallest”, “distinct positive”, “fixed sum”, “prove impossible”, “find the maximum”, or “equality occurs only when...”. These are signals to look for a boundary, an invariant, or a square.

How to recognise the method

If the expression is symmetric, try to write it as a sum of squares of differences. If the numbers are ordered, compare the extreme elements. If the condition holds for every pair or every partition, look for a global estimate: a sum, a product, or the number of positive/negative elements.

Typical mistakes

Do not multiply inequalities before checking signs. Do not forget equality cases. Another common mistake is estimating each term separately when the problem actually requires an extremal element or the whole set at once.

Mini-checklist

1. Where can equality occur? 2. Which expressions are certainly nonnegative? 3. Can the objects be ordered? 4. Is multiplying inequalities safe? 5. Is there an extremal element? 6. Are all impossible cases really excluded?

Key idea

AM-GM is useful not merely because positive numbers appear, but because one can choose terms whose product is controlled. The olympiad difficulty is not applying the formula, but preparing the expression for it.

Basic facts

For \(x_1,\ldots,x_n>0\), \(\frac{x_1+\cdots+x_n}{n}\ge\sqrt[n]{x_1\cdots x_n}\). Weighted form: if the weights are positive and sum to \(1\), then \(\lambda_1x_1+\cdots+\lambda_nx_n\ge x_1^{\lambda_1}\cdots x_n^{\lambda_n}\). Equality occurs when all terms chosen for AM-GM are equal.

When to use this method

Use it for fixed sums, fixed products, expressions like \(x+\frac{k}{x}\), products of several positive quantities, and maximum/minimum problems with an equality case.

How to recognise the method

Look for “positive numbers”, “sum equals”, “product equals”, “find the greatest/smallest”. If the expression does not fit immediately, try multiplying, dividing by a fixed quantity, ordering the variables, or substituting part of the condition.

Typical mistakes

Do not apply AM-GM to nonpositive quantities. Do not forget that equality requires equality of the exact terms to which AM-GM was applied. A common mistake is finding a correct bound without checking whether equality is compatible with the conditions.

Mini-checklist

1. Are all chosen terms positive? 2. Is their product known or bounded? 3. Is equality compatible with the condition? 4. Is a weighted form needed? 5. Is normalization or ordering needed first?

Key idea

Cauchy-Schwarz can replace a sum of fractions by one global fraction. In olympiad problems, the main task is choosing numerators and denominators so that the denominator sum becomes simple.

Basic facts

Standard form: \((a_1^2+\cdots+a_n^2)(b_1^2+\cdots+b_n^2)\ge(a_1b_1+\cdots+a_nb_n)^2\). Engel form: for \(q_i>0\), \[\frac{x_1^2}{q_1}+\cdots+\frac{x_n^2}{q_n}\ge\frac{(x_1+\cdots+x_n)^2}{q_1+\cdots+q_n}.\]

When to use this method

Use it for sums such as \(\frac{x^2}{A}+\frac{y^2}{B}+\frac{z^2}{C}\), fractions with cyclic denominators, reciprocal estimates, and problems where the sum of denominators is controlled by the condition.

How to recognise the method

If a numerator is a square or can be made into a square, try Engel form. If the numerator is not a square, rewrite \(\frac{a}{b}\) as \(\frac{(\sqrt{a})^2}{b}\), or multiply the fraction into a useful form.

Typical mistakes

Do not forget positivity of denominators. A common mistake is applying Cauchy with the wrong numerators and getting a useless denominator sum. Another mistake is losing the equality case.

Mini-checklist

1. Are all denominators positive? 2. Which numerators give the needed sum? 3. Does the denominator sum simplify? 4. Is a preliminary denominator estimate needed? 5. Is equality compatible with the condition?

Key idea

If two sequences are sorted in the same order, pairing them term by term gives the largest possible product sum. If they are sorted in opposite orders, the product sum is smallest. Chebyshev's inequality turns this ordering idea into an estimate for the average product.

Basic facts

Rearrangement inequality. If \(a_1\le\cdots\le a_n\) and \(b_1\le\cdots\le b_n\), then for any permutation \(\sigma\)

\[\sum_{i=1}^n a_i b_i\ge \sum_{i=1}^n a_i b_{\sigma(i)}\ge \sum_{i=1}^n a_i b_{n+1-i}.\]

Chebyshev's inequality. For similarly ordered sequences,

\[\frac1n\sum_{i=1}^n a_i b_i\ge \left(\frac1n\sum_{i=1}^n a_i\right)\left(\frac1n\sum_{i=1}^n b_i\right).\]

When to use this method

Use the method when a problem involves ordering, permutations, a maximum or minimum product sum, or power sums such as \(\sum a_i^{m+1}\), which can be viewed as products \(a_i^m\cdot a_i\).

How to recognise the method

Look for two monotone sequences: \(a_i\) and \(a_i^2\), \(a_i^2\) and \(a_i^3\), numbers and their reciprocals in the opposite order, or coefficients and variables with zero sum. If the expression changes when letters are permuted, rearrangement is often worth testing.

Typical mistakes

Do not apply the inequality before checking the order. Reciprocals have the opposite order. In symmetric problems one may rename variables first, but in cyclic expressions this must be done carefully: after renaming, the cyclic sum is only some permutation of products.

Mini-checklist

1. Which two sequences should be ordered? 2. Are they similarly or oppositely ordered? 3. Is the goal a maximum, a minimum, or an average estimate? 4. Can the cyclic sum be treated as an arbitrary permutation? 5. When can equality occur?

Key idea

Jensen's inequality says that for a convex function, the value at the average is not greater than the average value of the function; for a concave function, the direction reverses. In olympiad problems, the real skill is not naming convexity, but choosing the right function for the given constraint.

Basic facts

If \(f\) is convex, then

\[\frac{f(x_1)+\cdots+f(x_n)}{n}\ge f\left(\frac{x_1+\cdots+x_n}{n}\right).\]

If \(f\) is concave, the inequality reverses. Common functions: \(x^2\), \(x^4\), \(1/x\), \(e^x\) are convex on the relevant domains; \(\sqrt{x}\) and \(\ln x\) are concave for \(x>0\). A tangent to a convex function lies below the graph, while a tangent to a concave function lies above the graph.

When to use this method

Use the method with a fixed sum, an average value, products after taking logarithms, sums of roots, reciprocals, and problems where equality is expected at \(x_1=\cdots=x_n\).

How to recognise the method

Look for expressions of the form \(\sum f(x_i)\) and a condition on \(\sum x_i\). If a product must be estimated, take logarithms. If \(x^x\) appears, consider \(x\ln x\). If the problem asks for the maximum of a sum of roots or logarithms, concavity is likely involved.

Typical mistakes

Do not forget domains: \(\ln x\), \(\sqrt{x}\), and \(1/x\) require positivity or nonnegativity. A common mistake is reversing the inequality in the wrong direction for a concave function. Another mistake is applying Jensen when no fixed average is available.

Mini-checklist

1. Which function \(f\) is hidden in the expression? 2. Is it convex or concave on the required domain? 3. What is known about the average of the arguments? 4. Where should equality occur? 5. Is a logarithm or tangent line needed first?

Key idea

The UVW method works with symmetric inequalities in three variables. Instead of \(a,b,c\), we look at \(p=a+b+c\), \(q=ab+bc+ca\), \(r=abc\). Often, after fixing \(p,q\), the expression depends on \(r\) in a simple way, and the hardest case occurs on the boundary or when two variables are equal.

Basic facts

Main notation: \(p=a+b+c\), \(q=ab+bc+ca\), \(r=abc\). Useful formulas:

\[a^2+b^2+c^2=p^2-2q,\qquad a^3+b^3+c^3=p^3-3pq+3r.\]

Also \(\sum_{\mathrm{sym}}a^2b=pq-3r\), where \(\sum_{\mathrm{sym}}a^2b=a^2b+a^2c+b^2a+b^2c+c^2a+c^2b\). Schur degree \(3\): \[p^3-4pq+9r\ge0.\]

When to use this method

Use the method when the inequality is symmetric in \(a,b,c\), equality is expected at \(a=b=c\) or at two equal variables, and the expression can be rewritten through \(p,q,r\). It is especially useful for degrees \(3\) and \(4\), Schur-type inequalities, and fixed-sum problems.

How to recognise the method

Check whether the problem is unchanged by permuting \(a,b,c\). Then identify the degree and try to express the sums through \(p,q,r\). If after fixing \(p,q\) the dependence on \(r\) is linear, it is natural to check the cases \(b=c\) and \(c=0\).

Typical mistakes

Do not apply UVW to a nonsymmetric expression. Do not confuse the symmetric sum \(\sum_{\mathrm{sym}}\) with a cyclic sum. Another mistake is checking only \(a=b=c\): a proof usually requires the two-equal case or the boundary.

Mini-checklist

1. Is the inequality symmetric? 2. Which \(p,q,r\) appear? 3. Are \(p\) or \(q\) fixed? 4. Can Schur be applied? 5. What do the cases \(b=c\) and \(c=0\) give? 6. Does the equality case match the answer?

Key idea

A homogeneous inequality keeps its essential form when all variables are multiplied by the same positive number. Therefore we may choose a convenient normalization: \(a+b+c=1\), \(abc=1\), \(a+b+c=3\), or \(c=1\).

Basic facts

An expression has degree \(d\) if replacing \(a,b,c\) by \(ta,tb,tc\) multiplies it by \(t^d\). An inequality is homogeneous if both sides have the same degree. For example, \(\sum a^2\ge ab+bc+ca\) has degree \(2\), while \(\sum a/(b+c)\ge3/2\) has degree \(0\).

When to use this method

Normalization is useful when the expression is bulky but all terms have the same degree. Fix the sum for problems involving \(a+b+c\), fix the product for problems involving \(abc\), and fix one variable for two-variable or degree-zero fraction problems.

How to recognise the method

Check what happens under \(a,b,c\to ta,tb,tc\). If both sides are multiplied by the same factor, normalization is allowed. If a constant appears, make sure the expression has degree \(0\), or first homogenize the constant using \(a+b+c\), \(abc\), or another fixed quantity.

Typical mistakes

Do not set \(a+b+c=1\) in a nonhomogeneous inequality. Do not set \(abc=1\) if variables may be zero. Another mistake is forgetting to return to the original form after normalization or losing the equality case.

Mini-checklist

1. What is the degree of each side? 2. Are all variables positive? 3. Which normalization simplifies the expression? 4. What happens to equality? 5. Must a constant be homogenized first?

Key idea

A substitution is not decorative; it turns an expression into a more familiar form: removing ratios, encoding \(abc=1\), replacing triangle sides by sums, removing roots, or using the equality case.

Basic facts

For ratios, one often sets \(x=a/b\). If \(abc=1\), look for variables with product \(1\), or write \(a=x/y\), \(b=y/z\), \(c=z/x\). If \(a,b,c\) are triangle sides, use \(a=y+z\), \(b=z+x\), \(c=x+y\), where \(x,y,z>0\).

When to use this method

Use substitution when the condition is awkward, fractions contain repeated ratios, roots \(\sqrt{a}\) appear, triangle sides are involved, a constraint \(x^2+y^2=1\) appears, or equality is expected at \(a=b=c\).

How to recognise the method

If \(a/b,b/c,c/a\) appear, look for product \(1\). If \(b+c-a\), \(c+a-b\), \(a+b-c\) appear, the triangle substitution is probably needed. If the expression contains \(\sqrt{ab}\), try \(a=x^2\), \(b=y^2\).

Typical mistakes

Do not substitute without checking reversibility and domain: \(x,y,z\) must be positive where required. In the triangle substitution, remember \(x=(b+c-a)/2\), \(y=(c+a-b)/2\), \(z=(a+b-c)/2\). After solving, return to the original variables.

Mini-checklist

1. Which condition is awkward? 2. What becomes simpler after substitution? 3. Is the substitution reversible? 4. Is positivity preserved? 5. Where is the equality case?

Key idea

A constraint in an inequality usually tells us where to look for the extremum. With a fixed sum, expressions involving convex functions are often minimized at equal variables, while expressions involving concave functions are often maximized at equal variables. With a fixed product, AM-GM and the normalization \(abc=1\) are natural.

Basic facts

If \(a+b+c=S\), then \(\sum a^2\ge S^2/3\), \(ab+bc+ca\le S^2/3\), and \(abc\le(S/3)^3\). If \(abc=P\), then \(a+b+c\ge3\sqrt[3]{P}\) and \(ab+bc+ca\ge3\sqrt[3]{P^2}\). With fixed \(q=ab+bc+ca\), the relations \(p^2\ge3q\) and \(q^2\ge3pr\) are useful.

When to use this method

Use this approach when the condition fixes a sum, product, \(ab+bc+ca\), sum of squares, or average value. The problem often becomes an extremum problem for one symmetric quantity.

How to recognise the method

If equality is expected at \(a=b=c\), first try Jensen, AM-GM, Cauchy, or UVW. If variables are nonnegative and the sum is fixed, the maximum of some expressions may occur on the boundary: for example, \(\sum a^2\) is maximal at \(S,0,0\).

Typical mistakes

Do not confuse maximum and minimum: with a fixed sum, \(\sum a^2\) is minimized at equality but maximized on the boundary. Do not use \(abc=1\) if variables may be zero. Always check whether equality is attainable in the allowed domain.

Mini-checklist

1. What is fixed? 2. Are we looking for a maximum or a minimum? 3. Is equality at equal variables or on the boundary? 4. Is Jensen, AM-GM, Cauchy, or UVW needed? 5. Has the equality case been checked?

Key idea

A hard inequality problem is rarely solved by one standard formula. Usually two or three moves must be combined: normalization, Cauchy, AM-GM, Jensen, UVW, substitution, or a local estimate that later telescopes.

Basic facts

The main tools in this module are Engel form, AM-GM with chosen factors, Jensen for convex and concave functions, Schur/UVW for symmetric expressions, and SOS ideas: reducing a difference to a sum of squares or a product of nonnegative factors.

When to use this method

If a standard method gives a weak estimate, look for a preliminary substitution or an additional local estimate. If the expression is cyclic with roots, try proving one inequality for a pair of variables and summing it. If several numbers have distance conditions, use the sum of squared pairwise differences.

How to recognise the method

Symmetry suggests UVW; fractions with squares suggest Cauchy; fixed sum and roots suggest Jensen; products and powers suggest AM-GM; distance conditions suggest pairwise differences; expressions like \((x-y)\sqrt{\cdots}\) suggest comparing the root with a linear expression.

Typical mistakes

A common mistake is applying a strong theorem without checking the domain or direction. Another is obtaining a true but too weak estimate. In hard problems, it is important to know in advance what should telescope or where the equality case should appear.

Mini-checklist

1. Is there a normalization? 2. Which standard method gives the first step? 3. Where is the equality case hidden? 4. Is a local estimate needed? 5. Does the result sum to the required form? 6. Is the obtained estimate strong enough?

Key idea

In a mixed set, the statement no longer tells the student which method to use. The main task is to recognise the shape of the expression: fractions, powers, roots, symmetry, constraints, an extremal case, or a hidden identity.

Basic facts

This module uses AM-GM, Cauchy in Engel form, Holder, local estimates, Schur, identities for pairwise differences, and equality-case checking. Fractions with a manageable denominator sum often suggest Cauchy; cyclic cubes often suggest a local estimate; symmetric powers often suggest Schur or power means.

When to use this method

If denominators are sums, first try Cauchy. If the expression is homogeneous, check normalization. If \(a^3\), \(a^5\), and symmetry appear, look for Schur. If several numbers have distance conditions, sum the squares of all pairwise differences.

How to recognise the method

The phrase “for all real \(x\)” often leads to the discriminant. Roots of the form \(\sqrt{u^2+3v^2}\) can be compared with \(u+v\). Inequalities involving \(a+b+c\) and \(abc\) often need AM-GM or Schur. If equality is expected at \(a=b=c\), look for an estimate that becomes equality there.

Typical mistakes

Do not apply the strongest familiar method mechanically. A common mistake is getting a true but too weak estimate. Another is forgetting that multiplying by a negative factor reverses the inequality. In mixed problems, always check whether the equality case matches the statement.

Mini-checklist

1. Is the expression homogeneous? 2. Is there a natural equality case? 3. Can Cauchy be applied to the fractions? 4. Is a local estimate available for one term? 5. Is Schur or a power mean needed? 6. Is there a hidden identity for pairwise differences?

Key idea

A mock set tests not one topic, but the ability to choose a method. One set may contain Cauchy, AM-GM, Holder, Schur, local estimates, discriminants, and identities for pairwise differences.

Basic facts

Before starting a set, recall five basic moves: a fraction with a manageable denominator sum often uses Cauchy; products and fixed products use AM-GM; cubes in numerators suggest Holder or a local estimate; symmetric powers suggest Schur; “for all \(x\)” conditions suggest the discriminant.

When to use this method

If a problem looks short, first look for a standard move. If the standard move gives a weak estimate, try a local replacement or normalization. If many variables have distance constraints, use the sum of squared pairwise differences.

How to recognise the method

Equality at \(a=b=c\) often points to a symmetric estimate. Equality at boundary values often requires a constraint or boundary analysis. If the expression is homogeneous, one quantity can often be normalized; if it is not homogeneous, first look for a fixed sum or product.

Typical mistakes

Do not start with the hardest problem in the set. Do not use an inequality without checking direction and domain. Do not leave a solution at “obvious by AM-GM”: an olympiad solution should show exactly which terms are being compared.

Mini-checklist

1. Find the easiest entry point. 2. Mark the likely equality case. 3. Check Cauchy for fractions. 4. Check AM-GM for products. 5. Recall Schur for symmetric powers. 6. After solving, check whether the estimate was strong enough.