Maths WorkshopMaths problems with full solutions

Inequalities

67 problems · Algebraic inequalities, AM-GM, Cauchy-Schwarz and bounding techniques.

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Ex.179A classic inequality

#inequalities #AM-GM

Ex.191Squares and pairwise products

#inequalities #squares

Ex.489The AM-GM inequality

#AM-GM #inequality #arithmetic and geometric means

Ex.490The triangle inequality

#triangle inequality #absolute value #real numbers

Ex.610Minimising 1/a + 4/b + 9/c on a + b + c = 1

#Cauchy-Schwarz #constrained minimum #equality case

Ex.611Nesbitt's inequality

#Nesbitt #Cauchy-Schwarz #Titu's lemma

Ex.612Two inequalities from AM-GM on pairs

#AM-GM #symmetric inequalities #warm-up

Ex.613a²/b + b²/c + c²/a ≥ a + b + c

#Titu's lemma #Cauchy-Schwarz #cyclic sums

Ex.614a³ + b³ + c³ ≥ a²b + b²c + c²a

#rearrangement inequality #AM-GM #cyclic sums

Ex.615Weighted AM-GM: how to choose the splitting

#weighted AM-GM #optimisation #equality case

Ex.642A chain of squares: a⁴ + b⁴ + c⁴ ≥ abc(a + b + c)

#sum of squares #SOS #real numbers

Ex.699AM-GM for three numbers

#inequalities #AM-GM

Ex.1011AM-GM for three numbers

#AM-GM #three variables #inequality

Ex.1012The arithmetic-harmonic mean inequality

#AM-HM #harmonic mean #inequality

Ex.1013Bernoulli's inequality

#Bernoulli #inequality #induction

Ex.1014Cauchy-Schwarz by induction

#Cauchy-Schwarz #induction #inequality

Ex.1015Cauchy-Schwarz inequality

#Cauchy-Schwarz #inequality #sequences

Ex.1016Geometric mean and harmonic mean

#geometric mean #harmonic mean #inequality

Ex.1017Jensen's inequality for the square function

#Jensen #convexity #square function

Ex.1018Discriminant and positivity

#discriminant #positivity #quadratic

Ex.1019The rearrangement inequality

#rearrangement #inequality #sequences

Ex.1020Titu's lemma (Cauchy in Engel form)

#Titu #Engel form #Cauchy-Schwarz

Ex.1265Chebyshev's sum inequality and a^a b^b c^c

#Chebyshev's sum inequality #logarithm #monotone sequences

Ex.1266A sum of squares of a + 1/a, via Jensen

#Jensen's inequality #convexity #constraint a+b+c=1

Ex.1267The tangent line trick

#tangent line trick #constraint a+b+c=1 #one-variable reduction

Ex.1301P′² ≥ P·P″ for polynomials with real roots

#polynomials #real roots #logarithmic derivative #inequalities

Ex.1322A symmetrisation trick: Σ a³/(a² + ab + b²)

#symmetrisation #cyclic sums #tangent-style bound

Ex.1323IMO 1984: yz + zx + xy − 2xyz

#IMO 1984 #pqr method #case analysis #AM-GM

Ex.1324The product of the sines of a triangle

#Jensen #concavity #logarithm #triangle

Ex.1325Minkowski's inequality and a sum of two square roots

#Minkowski #triangle inequality #optimisation

Ex.1326Muirhead's idea with weighted AM-GM

#Muirhead #majorization #weighted AM-GM

Ex.1327Ravi substitution: abc ≥ (a + b − c)(b + c − a)(c + a − b)

#Ravi substitution #triangle inequality #AM-GM

Ex.1731The weighted AM-GM inequality

#weighted AM-GM #inequality #convexity

Ex.1732Cauchy-Schwarz and Jensen

#Cauchy-Schwarz #Jensen #convexity

Ex.1733Chebyshev's sum inequality

#Chebyshev #sum inequality #monotone sequences

Ex.1734Nesbitt's inequality

#Nesbitt #inequality #fractions

Ex.1735Nesbitt's inequality via Cauchy-Schwarz

#Nesbitt #Cauchy-Schwarz #Engel form

Ex.1736Power mean inequality

#power mean #inequalities #means

Ex.1737Positivity of a quadratic expression

#positivity #discriminant #inequalities

Ex.1738Bounding a sum of square roots

#square roots #AM-GM #inequalities

Ex.1739A ratio-sum inequality

#ratios #AM-GM #cyclic sums

Ex.1740An inequality by substitution

#substitution #AM-GM #inequalities

Ex.1741Sum of square roots

#square roots #Cauchy-Schwarz #inequalities

Ex.1742Cauchy-Schwarz via sums of squares

#Cauchy-Schwarz #sums of squares #algebraic proof

Ex.1743Cyclic sum of squares

#cyclic sum #squares #factorization

Ex.1744Minkowski's inequality

#Minkowski #triangle inequality #norms

Ex.1935A polynomial with all roots real and Vieta

#Vieta #discriminant #real roots #Newton inequality

Ex.1950A classic with abc = 1 (IMO 1995)

#IMO #substitution #Titu's lemma #abc=1

Ex.1951Mixing variables: a²b + b²c + c²a ≤ 4/27

#mixing variables #cyclic sums #ordering #AM-GM

Ex.1952Schur's inequality and a bound with a + b + c = 1

#Schur's inequality #IMO 1984 #symmetric polynomials #ordering variables

Ex.1990A cyclic inequality: a²b + b²c + c²a + abc ≤ 4

#cyclic inequality #WLOG ordering #AM-GM

Ex.1991With abc = 1: Σ 1/(a² + a + 1) ≥ 1

#constraint abc=1 #clearing denominators #expansion

Ex.1992Hölder's inequality for three sequences

#Hölder #AM-GM #normalisation

Ex.1993Two inequalities by induction on the number of variables

#induction #AM-GM #Bernoulli #smoothing

Ex.1994The cube substitution: Σ 1/(a + b + 1) ≤ 1

#substitution #abc=1 #cubes #AM-GM

Ex.2158Cyclic SOS decomposition

#SOS #cyclic #symmetric

Ex.2159Hölder's inequality

#Hölder #inequality #conjugate exponents

Ex.2160A direct application of Hölder

#Hölder #powers #inequalities

Ex.2161Muirhead's inequality

#Muirhead #symmetric inequalities #majorization

Ex.2162Generalized Nesbitt inequality

#Nesbitt #generalization #n variables

Ex.2163Schur's inequality

#Schur #symmetric inequalities #factorization

Ex.2164Sum of squares decomposition

#SOS #sum of squares #factorization

Ex.2165The sum-of-squares method

#SOS #sum of squares #symmetric inequalities

Ex.2166Schur's inequality via SOS

#Schur #SOS #symmetric

Ex.2167The uvw method

#uvw #symmetric inequalities #pqr

Ex.2231IMO 2001: a/√(a² + 8bc)

#IMO #weighted Jensen #convexity #normalisation

Ex.2232Vasc's inequality by a sum of squares

#SOS #sum of squares #cyclic inequality #Vasc's inequality